Duplicate bridge certainly has full-time professional players who earn their living playing, although most of them are also teachers, and play with students for pay. There at least used to be professional players of Chemin de fer (a gambling game) and still are for blackjack and other casino gambling games.
I won't give a list since a list will keep changing. Rather, I'll give a method to find these games.
Go to https://www.twitch.tv/, which is the most popular platform used by streamers to stream themselves playing games. Go to the categories section, and sort all categories by viewer count. If we agree that the games most likely to have full-time professional ...
There are a number of Magic: the Gathering players who derive all their income from the game, whether that be:
A direct contract from Wizards of the Coast, given to the players with the best competitive results
Prize money from winning tournaments
Writing about the game
Sponsorship related to all the above
The split ...