Price seems to still have his M1911 in the end of Kingfish. However, in the level "Game Over!" of the first game, he gives it to Soap and only gets it back in "The Gulag" in part two.
If you look closely, the pistol Price uses in Kingfish looks more like a SIG P226 or similar than an M1911. So if it is canon, Soap simply never gave Price his pistol back until "The Gulag", for some reason.