Quiddich?
Paintball 
Did you even read the OP?
Wasn’t there a variant of 43-man-squamish with multiple balls in play at once? If there wasn’t there should have been.
When I was in primary school, in addition to multi-ball dodgeball, we also played multi-ball four square.
Quidditch started as a fictional game but there are real people who play a non-flying adaptation that they call Muggle Quidditch. Some people are even advocating for it to be a recognized intermural NCAA sport: A Muggle's Dream: Quidditch As A NCAA Sport : NPR
You know, that might actually make soccer interesting.
You don’t want to eat the cookie do you?
Croquet always has two balls in play, or else it wouldn’t be much of a game. Who would you be playing against?
I meant that in croquet, like in golf or billiards, don’t you always just hit one ball on your turn? There are never two balls being struck at the same time by different players.
You have more than one ball in play at a time? Not that I’ve seen. Pachinko might count, but not (IMO) traditional pinball.
On some machines if you hit a certain target or combination of targets, two extra balls are released into play. Wackiness ensues.
Also, on some very old machines you can pump more than one ball onto the field at once.
There are plenty of games with multiball action, going back over decades of pinball play.
That’s a great idea from the OP. I think I’ll try that out.
Leave it to the French to come up with this… 
However, your opponent’s ball is in play, because you can hit your opponent’s ball with your ball. I don’t think that’s allowed in golf, for example, though there probably is a rule about accidentally hitting your opponent’s ball.
Similarly, in lawn bowls there can be several of your balls and of your opponents’ balls in play, though you can only roll one ball at a time. Again, as in croquet, you can deliberately hit any of the balls in play on the green.
There’s a rule for everything in golf. If you hit someone else’s ball, you put it back to its original location the best that you can. You play your ball wherever it stopped. There is no penalty.