(This seems like the kind of idea that would already be at least half-baked. I found skeet golf, and golf using paintballs as balls, but this seems significantly different from both.)
Players play as per usual - on their own shots. Their opponents fire paintballs at the ball. For foursomes, each
player uses a different colour.
Rules are as for normal golf except:
1. After coming to rest, the ball is examined for splodges. For each hit, two strokes are added to the player's score for the hole. Each opponent can only score 1 hit per shot, though (e.g. 2 red and 1 yellow splodges = 2 extra strokes).
2. Players can choose to declare their ball 'lost' and take a drop shot (worthwhile if they think it might've been hit by several players).
3. Hits on putts count as one extra stroke rather than two. Use the broom beside the green to smooth paintmarks after playing the hole.
4. No extra strokes are added on shots which are holed.
5. Opponents must stand a minimum distance from the player, and except for putts, opponents must all stand to the same side of the player to minimise friendly fire risks. (Though combat gear would make a nice change to normal golf attaire.) Players must wait for opponents to take their position before playing their shot.
6. Opponents who fire before the ball is struck incur a one stroke penalty. Players are allowed to 'dummy' a maximium of twice before playing each shot.
7. The number of paintballs for the round is limited to (e.g.) 500 per player per opponent (so that shooters can't just blast away crazily on every shot (unless you want to play it that way ;-) )). (If it turns out to be too easy to hit putts, a further restriction could be placed e.g. one shot per putt.)
Restrictions on the number of paintballs would also add extra strategy in foursomes - three players might choose to gang up and use their shots against the player currently in the lead.
If you like, I can throw in some bonuses (e.g. birdieing a hole allows you use of a more powerful gun and bonus balls on the next hole). But I think this could work well as it is. Sure, it might be hard to hit the balls, but even a few more hits than your opponent could be enough to win you the round.
Also, XTreme Paintgolfball (using the rules of XTreme Golf) could work well, as you'd have to decide whether to use up time trying to hit opponents' balls, maybe even lying in wait for them to play their shot, or to dash on ahead of them. In this version you'd be allowed to hit their ball at any time it is in motion, from any position.