The mile was not standardized in England until 1592, and even then it was only the English mile (different distances prevailed in Scotland and Ireland). Outside of Britain the international recognition of the mile as a standard distance waited on the spread of the British Empire to Africa and Asia (let's say roughly later than 1800). So the question narrows down to whether any competitor in Britain or North America broke the mile record prior to that time.
Timing is another issue, accurate portable (spring-driven) clocks were not available until well after 1759 (invention of the Lever escapement by Thomas Mudge). The most accurate clock at that time was Harrison's clock (a pendulum clock used for navigation), good to 3 seconds a day, but only a small handful of those clocks were in existence at that time for the sole purpose of allowing ships to determine longitude at sea.
So we can eliminate any claims to a measurable four minute mile prior to 1760 (at best)
We also have to consider that hand timing invariably reduces the athlete's elapsed time by about .18 +- .08 seconds in scientific studies of hand timing of 40 yard dashes. Perhaps we should also ask the question, did Bannister really break 4 minutes - one might think that the hand timing of a mile race would have significantly greater error than the timing of a 40 yard dash? I'm not sure 3:59.4 gives him a big enough margin of error.