Kirby was born in Wimbledon, London on 18 April 1988, the daughter of former Country Living magazine editor Jane and urologist Roger Kirby. She has two siblings, Joe and Juliet. Actors Vanessa Redgrave and Corin Redgrave were family friends. She was educated at Lady Eleanor Holles School in Hampton. After being turned down by Bristol Old Vic Theatre School, she took a gap year to travel before studying English at the University of Exeter.

She then went on to appear at the National Theatre as Isabella in Women Beware Women by Thomas Middleton, directed by Marianne Elliott, alongside Harriet Walter and Harry Potter’s Harry Melling. She then starred as Rosalind in As You Like It by William Shakespeare at the West Yorkshire Playhouse in Leeds; Alfred Hickling of The Guardian described her as a “significant new talent”. In 2011, she was in the première of The Acid Test by Anya Reiss at the Royal Court Theatre, directed by Simon Godwin, earning praise for her performance from Paul Taylor of The Independent, who described her as “a star if ever I saw one”.

In 2011, Kirby made her television debut in the BBC’s The Hour, alongside Ben Whishaw, Dominic West, and Romola Garai. She played Estella in the BBC’s mini-series adaptation of Great Expectations alongside Ray Winstone, Gillian Anderson, and Douglas Booth.

She went on to play Masha in the acclaimed production by Benedict Andrews of Three Sisters at the Young Vic in September 2012, earning exceptionally good reviews, with Matt Trueman of Time Out stating: “In a super cast given licence to shine, Kirby stands out as Masha”.

She filmed The Rise in early 2012 alongside Matthew Lewis and Timothy Spall. The film premièred at the Toronto and London Film Festivals to favorable reviews, and won the Best Debut Category for director Rowan Athale.

In 2013, Kirby returned to the National Theatre to play Queen Isabella in Edward II opposite John Heffernan. In the summer of 2014, she played Stella in A Streetcar Named Desire, again collaborating with Benedict Andrews at the Young Vic, alongside Gillian Anderson as Blanche and Ben Foster as Stanley. She won Best Supporting Actress category at the Whatsonstage Awards 2014, which is voted for by the public.

In 2015, she appeared in Everest as American socialite Sandy Hill Pittman, The Dresser, and in May of that year, was cast as Princess Margaret in Netflix’s first original British series The Crown. Her selection came after a six-month search. For this role, she was nominated for a BAFTA award in 2017, and won the award for season two in 2018. In 2020, Kirby won the Best Actress Award at the 2020 Venice Film Festival for Kornél Mundruzcó’s English debut Pieces of a Woman, a film portraying the degradation of a marriage.

Kirby played Elena in Robert Icke’s production of Uncle Vanya at the Almeida Theatre in 2016, for which she won rave reviews, with Matt Trueman of Variety writing: “Kirby is a superlative Elena: lithe, fickle, hypocritical, shallow and yet always sympathetic. It’s a performance that confirms her as the outstanding stage actress of her generation, capable of the most unexpected choices.”

In 2018, Kirby played the title character in Polly Stenham’s Julie, an adaptation of August Strindberg’s Miss Julie, at the National Theatre.

Kirby was in a long-term relationship with actor Callum Turner, with whom she co-starred in Queen & Country. The relationship ended in February 2020.