Her breakthrough role was playing Joe as a young woman in Lars von Trier's 2013 drama film Nymphomaniac.

Other credits include The Childhood of a Leader (2015), All the Money in the World (2017), Amanda (2018), The Evening Hour (2020), Lovers (2020), The Serpent (2021), and The Brutalist (2024).

She moved to Tokyo when she was seven, where she lived with her French father, René Martin, a hairstylist, and her English mother, Annette, until she was thirteen.

[4][5] For her role in the film, which featured unsimulated sex, Martin had a "porn double" and used a prosthetic vagina.

She was also a part of the Breakthrough Brits of 2014 which was organized by BAFTA to recognize emerging talents.