Dodgeball: URF mode.
In all seriousness, though, you can always turn to FPS games for inspiration. You can try some CTF or control points-style thing, where you have to get a certain number of balls to a point to win. This splits their goals between getting the other team out to get more balls, or saving balls to put on the point. RiotShadowStorm mentioned an Infection style game: another version is a FFA type where the zombies can only tag people, where others can use dodgeballs.
Alternatively, you can try some other games, kickball for instance. I used to play a variation called Alaskan Kickball (or maybe it was Australian, I forget), where you play on a basketball court and instead of tagging someone with the ball to get them out, you had to score a basket.
We also played a game of CTF tag, where the field was split into halves, and if you got tagged (two hands) on your opponent's side, you were out and had to go sit in their 'jail' way in the back of their base. You could free your teammates by running all the way through the enemy base and tagging a person in jail. Also the area immediately around the flag and jail were safe zones once you made it there, allowing you to prep a sprint back to your own team.