Language is the key to communication. But it's also the key to miscommunication, as Tim Cowbury's intense and supremely smart play The Claim demonstrates. In it, a man who is trying to claim asylum is interviewed, but though there are two people listening and questioning – one of whom speaks his own tongue – his words are twisted and contorted. They are shaped into a story that isn't his.

There's no before or after The Claim, no backstory, no easy narrative. In front of us suddenly is Serge and he is talking to someone else about sweets, about Streatham, about elephants. It all feels very genial and it could be two friends chatting, except for the tension in the air and the slight confusion. Serge tries to tell a story about Willy Wonka. His questioner thinks he's talking about the cartoon Willy Fog. Foggy it absolutely is.

Serge and his story get lost in translation and as the play progresses and a second interviewer joins the first (they are called only A and B) things take a darker turn. Serge slowly realises he must make sure he is understood if he wants to come out on top. But he begins to suspect that he isn't. Indeed, it becomes more and more frustrating to watch as he delivers his story and the people either side of him repeat it back. They change elements, interrogate the wrong moments as the words travel from Serge in French to A, from A to B in English. The story comes back to Serge reformulated so he gets increasingly agitated. Everything in the play is spoken in English, so that you can hear it all happening word for word.

If the play feels a little like you're watching a puzzle that's missing pieces, it's supposed to. It replicates what Cowbury is doing with the dialogue – confusion is king here. The stakes are slowly raised: what was originally a chat becomes the thing that will keep a person safe or put them at risk. So much, we begin to realise, pivots on this one conversation and no one seems to be getting it right.

Ncuti Gatwa, Yusra Warsama and Nick Blakeley are all exceptionally poised as Serge, B and A and deal with the fast pace and key timing of the script beautifully. Warsama's subtle hand movements, her regimented, forced smiles demonstrate the awful rigidity of those enforcing rules, while Gatwa's wide eyed smiles at the beginning paint a good willed, innocent Serge: not someone who warrants the suspicion with which B treats him. Blakeley is a good humoured fool, hoping to help but not managing to really listen, he's too caught up in his own personal issues to be able to focus on another's.

The Claim is a masterclass in the pitfalls of language, a taut, funny but ultimately massively unsettling piece of work which is a stark and vital reminder of interpretation, viewpoint and how our black and white systems don't make enough room for vulnerable people. How would any of us cope at making ourselves clearly understood in a language we don't even know?

The Claim runs at Shoreditch Town Hall until 26 January.