Here's a bizarre scenario: a group called Tommy McCook and the Supersonics released a track called "Our Man Flint" on Jamaican labe Treasure Isle, which was owned by Duke Reid, a larger-than-life record producer. Wikipedia describes him thusly:
" He was dressed in a long ermine cloak and a gilt crown on his head,
with a pair of Colt 45s in cowboy holsters, a cartridge belt strapped
across his chest and a loaded shotgun over his shoulder. It was not
uncommon for things to get out of hand and it was said that Duke Reid
would bring the crowd under control by firing his shotgun in the air."
Research shows the record was released in 1968. As for this track, aside from McCook, a well-known jazz artist, shouting out "Our Man Flint!" at the beginning of the song, there is no reference to the main theme from the film. In fact, as one YouTube commentator points out, the song on the record is actually "Sway", a lounge hit for Dean Martin. The record label also doesn't list the requisite credits for the composer and songwriter. However, it is a good, funky version of the song.