Although born in France, Bamba represents his parents’ nation of Ivory Coast. He featured in the 2003 FIFA World Youth Championship, the 2005 African Youth Championship and the 2008 Toulon Tournament. Bamba was part of the Ivory Coast squad that reached the quarter-finals of the 2008 Olympic Games.

After an initial trial period, Bamba signed for Scottish Premier League side Dunfermline Athletic in July 2006 on a two-year contract. Jim Leishman later revealed that he had been intending to bring over another player from France on trial, but this deal fell through and Bamba’s agent sent him instead. Leishman was immediately impressed by Bamba’s physique and his performance as a trialist during a preseason friendly against West Bromwich Albion striker John Hartson. He made his competitive debut for the club in Dunfermline’s opening league match of the season, a 2–1 defeat to Hearts. On 2 December 2006, Bamba was sent off for the first time in his professional career after pushing Steven Naismith in the chest during a 5–1 defeat to Kilmarnock.

Bamba was allowed to go on trial with Championship side Watford in July 2008, appearing in a friendly for them against Boreham Wood. Soon afterwards, manager Jim McIntyre said he expected a fee to be agreed between Dunfermline and Watford, which would allow Bamba to move to the Championship side. The deal was thought to be almost complete, with Watford sending a doctor to China, where Bamba had travelled with the Ivory Coast Olympic squad, to conduct his medical. However, Watford later pulled out for unknown reasons. A few weeks later, Dunfermline accepted offers of £50,000 for Bamba to move to either Hibernian and Motherwell, keeping him in the Scottish Premier League.

Bamba chose to move to Hibernian, signing a three-year contract with the club. As part of the transfer deal, Dunfermline were due 10% of any future transfer fee. He was sent off on his debut for a second bookable offence against Dundee United, but Bamba subsequently established himself in the Hibs first team. Despite being a natural centre back, manager Mixu Paatelainen praised Bamba’s performance as a defensive midfielder during a 2–0 win over Celtic in December 2008. Paatelainen commented further that he believed Bamba had improved as a player since his spell at Dunfermline, with him no longer going to ground when making a challenge, and described him as possessing “tremendous pace and physical presence.” Bamba quickly became a popular player with the Hibs support due to his tough and committed tackling in this new role.

Bamba won his first cap at full international level for Ivory Coast in a 2008 friendly match against Israel, and his second cap as a late substitute in a friendly against Turkey. Bamba made his competitive debut in a 5–0 win against Malawi in a 2010 FIFA World Cup qualifying match, again as a late substitute. This match was played in the aftermath of the Houphouët-Boigny Arena stampede, which killed dozens of Ivorian spectators. Bamba became a regular in the national team as they qualified for the 2010 Africa Cup of Nations and the 2010 FIFA World Cup. Bamba scored his first goal for the team in a friendly against Rwanda, as part of the squad’s preparation for the 2010 African Cup. He played in all of Ivory Coast’s matches at the 2010 Africa Cup of Nations, but the team suffered a quarter-final defeat to Algeria after extra time. The “uneasy” central defensive partnership between Kolo Touré and Bamba was cited as one of the reasons for their defeat. The tournament had been played in the shadow of the shooting of the Togo team bus, however, with death threats also being made against the Ivory Coast team.

Following Paatelainen’s departure, new manager John Hughes moved Bamba back to a central defensive role after the departure of Rob Jones in the 2009 season. In his second match in this position, Bamba scored his first ever professional goal in a 3–1 win against Falkirk on 22 August, heading in from a John Rankin corner. Coincidentally, he scored his second goal for the club at the same ground against the same team and in another 3–1 win, on 27 March. The Edinburgh Evening News reported on 8 May 2010 that Bamba’s performances had “not gone unnoticed”, with scouts from Fulham and Rennes due to watch Bamba. Sven-Göran Eriksson, the new manager of the Ivorian national team, was also due to watch Bamba ahead of the 2010 World Cup, but was not spotted at the last game played by Hibs in the 2009–10 season.

Bamba angered manager John Hughes when he failed to report for a pre-season tour of the Netherlands in July 2010. Bamba responded angrily to the criticism by claiming that he needed a full three weeks off after returning from the World Cup and had attempted to contact both Hughes and Hibernian’s chairman to discuss the situation, but Hughes dropped Bamba from the first team squad when he eventually returned to Edinburgh. Bamba was made to train with the club’s youth team on his return but was recalled for the second leg of a UEFA Europa League tie with NK Maribor at Easter Road, which Hibs lost 3–2 on the night and 6–2 on aggregate. Following the incident, manager Hughes stated his desire for the situation to be forgotten. During his time with Hibs, he made 80 appearances in all competitions, scoring four times.

With Bamba’s contract with Hibs due to expire at the end of the 2010–11 season, the club agreed on a deal with Football League Championship club Leicester City for his transfer. The deal was confirmed on 2 January 2011, with Bamba joining for an undisclosed fee midway through the season, although sources placed the figure around £250,000 with Hibs also due to receive a percentage of any future transfer fee. He was reunited at Leicester with his former Ivory Coast manager, Sven-Göran Eriksson. He made his debut for the club in a 2–2 draw against Manchester City on 9 January in the third round of the FA Cup, scoring with his first touch of the ball. He scored his first two league goals in a 4–2 win over Millwall on 22 January.

After speculation linking him with several Premier League sides, including Blackburn Rovers, and French side Marseille during the 2011 summer transfer window, Bamba signed a new contract with Leicester in September 2011. During negotiations, Eriksson compared Bamba’s style of play to that of former German international player Franz Beckenbauer. Blackburn manager Steve Kean had claimed that the two clubs had made a bid for Bamba and described the move as being “quite close” to being completed.

Jim Leishman, who signed Bamba at Dunfermline Athletic, said of his former charge in 2010: “He’s certainly got the physique and he’s got the potential to do it. But potential sometimes passes you by, it’s all about going out and doing it. He’s a great runner, he’s as strong as an ox, and he’s a strong tackler.” Sky Sports’ Soriebah Kajue in September 2011 said Bamba “established himself as a rock at the heart of the Foxes defence and has produced a level of consistency this year that has simply been outstanding”, paying tribute to “Bamba’s burgeoning reputation as a ball-playing defender”.

Bamba signed for Turkish club Trabzonspor in June 2012 for a fee of £750,000 after being recommended to the club by his international teammate Didier Zokora. He made his debut for the club on 26 August 2012 playing the full 90 minutes, in a 2–0 victory at home to Elazigspor. He only made 18 appearances during his first season at the club, missing a proportion of fixtures in December and January in part because he was away with Ivory Coast as they qualified for the 2013 African Cup of Nations.

Bamba resumed his partnership with Touré in the 2012 Africa Cup of Nations. Ivory Coast did not concede a goal during the entire tournament, but lost the final against Zambia on a penalty shootout. At the 2014 FIFA World Cup, Bamba played in central defence alongside Zokora after Toure contracted malaria in the period before the tournament. Bamba was booked in the first match, which Ivory Coast won 2–1 against Japan. Ivory Coast were eliminated at the group stage after suffering 2–1 defeats against both Colombia and Greece.

During the 2013–14 season, Bamba made 15 appearances for Trabzonspor in the first half of the season, including being a regular starter in their Europa League campaign. However, on 22 January 2014, he picked up an ankle injury which ruled him out for the majority of the latter part of the season. Bamba was not selected to play for Tranzonspor again after suffering the injury, however he recovered in time to play for Ivory Coast at the 2014 FIFA World Cup.

On 26 August 2014, after having been released by Trabzonspor, Bamba agreed a three-year deal with newly promoted Serie A club Palermo. He made his Palermo debut on 24 September 2014, when he started the match for Palermo in a 3–3 draw against Napoli. However, he made no further appearances for the club. After leaving the club, he expressed his confusion over not playing more for Palermo, commenting “The manager would say ‘you’re not far off, keep doing what you’re doing and you’ll be playing.’ Well, after one month then two, three and four you start to get worried.”

Bamba was called up to the squad for 10 September 2014 fixture against Cameroon, however on 29 December 2014 he was a surprising omission from the Ivory Coast squad for the 2015 Africa Cup of Nations. Although he has not retired from international football, he has not appeared for the side since 2014.

On 23 January 2015, less than six months into his Palermo contract, Bamba was loaned to Championship side Leeds United for the remainder of the season, with the possibility of making the transfer permanent at a later date. He made his debut for Leeds in a 2–1 West Yorkshire derby win over Huddersfield Town a week later. Less than a month after joining the side, Bamba was named captain for three games in place of Liam Cooper. In particular, he served as captain when Leeds beat the then league leaders Middlesbrough 1–0, with Bamba receiving praise for his performance.

Nonetheless, new head coach Garry Monk announced Bamba would remain as Leeds captain for the 2016–17 season. On 7 August, Bamba scored an own goal against Queens Park Rangers in a 3–0 defeat in the opening match of the season and was subsequently dropped from the side, falling behind Kyle Bartley, Liam Cooper and Pontus Jansson in the pecking order. On 1 September 2016, Bamba, despite being Leeds’ captain, was released by the club. His contract was cancelled by mutual consent following the close of the transfer window the previous day. The decision was made after Monk informed him that he would no longer be first choice at the club.

Following his release, Bamba trained with Derby County who were managed by his former Leicester boss Nigel Pearson but was not offered a deal. He instead joined Neil Warnock at Cardiff City in October 2016, signing a two-year contract. Warnock had long been keen on signing Bamba at previous clubs and had contacted him a month before the move, when both were without a club, and urged him to wait for Warnock’s appointment at a new club to join him there. He made his debut for the club in a Severnside derby match against Bristol City on 14 October, scoring the winning goal as the Bluebirds won 2–1. His performances for Cardiff led manager Warnock to declare that “I wouldn’t want anyone else. I don’t think there’s anyone better.” Bamba received his first red card in seven years against Ipswich Town in December, following a scuffle with forward, David McGoldrick. After receiving the red card, Bamba clashed with the fourth official before being physically restrained by Warnock and leaving the pitch. His reaction led to a three match ban. In April, Bamba suffered a groin injury against Barnsley, which ended his season prematurely.

Bamba returned to the squad at the beginning of the 2017–18 season, helping Cardiff win their three opening games of the season, for the first time in their history. He signed a new three-year contract on 25 August 2017, following it up by heading in the winner against Queens Park Rangers the next day. He went on to help the club win promotion to the Premier League after finishing the season in second place. He was ever present in the league, appearing in all 46 league matches, and was named in the PFA Championship Team of the Year.

Sven-Göran Eriksson, who managed Bamba at club level for Leicester City and internationally for the Ivory Coast, compared the defender with Franz Beckenbauer. Cardiff manager Neil Warnock regularly told Bamba “Don’t be Beckenbauer. Just kick or head it. When it’s on, pass it, but if it’s not on, just put it up there,” and in December 2017 Warnock branded Bamba and fellow Cardiff centre-half Sean Morrison as better than Virgil van Dijk. Bamba’s favourite players growing up were Lilian Thuram and George Weah.

He made his debut in the Premier League in Cardiff’s first match of the 2018–19 season, a 2–0 defeat to A.F.C. Bournemouth, before scoring his first goal in the top division in a 4–1 defeat to Chelsea in September 2018. Bamba scored a further 3 goals during the season, including in wins against Brighton & Hove Albion and Southampton. Having missed only one match during the course of the season, he was ruled out for the remainder of the campaign after rupturing his cruciate knee ligaments in a defeat against Wolverhampton Wanderers on 5 March 2019.

In July 2019 it was revealed that Bamba would miss the start of the 2019–20 season due to injury.