Rugby / American Football
The essential problem with having very large open areas is that once a player launches themselves out into space, there is only so much they can do to change their trajectory; if they have the ball, they can throw it, but otherwise they can't do much.
That isn't necessarily a problem, and in fact could contribute to the style of the sport. For example, one could play a game similar to Rugby or American Football, except counting a 'down' or 'tackle' as soon as a defending player tags the player carrying the ball. Play would consist of players launching themselves on trajectories to intercept other players; if the player with the ball was going to be tackled, they would attempt to pass it to another player. The defending team would also attempt to predict the trajectory of and intercept the receiving player after they catch it.
Polo
One way to overcome the fixed-trajectory problem is to give the players some sort of vehicle, such as a jetpack, or have them ride on some type of vehicle, but this has safety issues due to the potential of high-speed midair collisions. However, this could be dealt with in the same way as modern-day Polo.
Polo is a game played by teams of players on horseback, with a set of rules dictating how players are allowed to move to ensure that riders and their horses do not collide in a way that would be dangerous. 732
Players use long mallets to strike a ball that rolls along the ground. In space, these might be replaced with some sort of racket, to make it easier to hit.
Ball Passing Games
(Basketball, Hockey, Lacrosse, Football, Handball, Water Polo, etc.)
Another way to solve the mobility problem would be to use some type of rope lattice to fill the arena. The rope would be thick enough and elastic enough that colliding with it would not be very likely to cause injury. With a lattice pitch of around 1.5-2 meters, it would be easy enough to reach out with one's hands or feet and hook a rope in order to change trajectory or come to a stop, and one could launch off the ropes easily enough. If necessary, padded columns could be added to give more support to the lattice, so that it wouldn't be able to stretch too far.
This could accommodate a large number of different games, using different implements to launch different sorts of projectiles, with different rules about moving and scoring. 214