Born in 1988, Kirby is the daughter of a magazine editor and a surgeon. She studied English at Exeter University.
Kirby starred as Estella in the BBC adaptation of Great Expectations in 2011, and as Joanna in Richard Curtis' romantic comedy About Time in 2013.  In 2018, Kirby was nominated for her an Emmy for her portrayal of The Princess. Kirby stars in the Strindberg play Julie at the London National Theatre, which runs 31 May—9 September 2018.