Is it permissible for one of the players to offer a prize (a certain number of bananas) to the winner of the game, provided that the prize giver is an "outsider" to that game, and then the players take turns offering the prize each time? And does the prize giver's participation in the competition change the ruling?

The meaning of gambling does not disappear if the contestants stipulate that one of them must provide what the winner takes each time. The risk still exists, unlike if a person who is not one of the contestants donates the prize, or if one of the contestants offers it without the others paying anything, or if everyone shares in the prize and includes someone who did not pay anything for it (the muhallil). These last scenarios are considered permissible by the majority of scholars.