Bannister has the official recognition.
But others have made claims with varying degree of proof.
Cunningham claimed to have done it in practice, but he never duplicated in competition, and his best official effort casts doubt on his being able to go sub 4.
Landy also claimed to have gone sub 4 before Bannister, and he backed it up with a 3:57 in actual competition, so his claim of going sub 4 before Bannister is more believable.
Big Hawk Chief has a claim for 3:57 from 1876. He did not push the claim himself, it is noted in passing in a US cavalry officers diary. It was not a publicity stunt and only came to light years after the fact, so deception is unlikely.
A few documented stories from even earlier show that a few athletes were within 10 seconds of going sub 4, so it is possible someone else went under 4 without a record of the event having survived.
Officially, Bannister is the man. But there will always be some doubt.