Ice hockey: does play stop when there's a fight?

In ice hockey, be it the NHL or another league, is play automatically stopped if a fight breaks out? I mean, do the officials stop play and pick up the puck, or could the other players continue to play?

Bonus question: Have teammates ever fought each other on the ice during a game, at any level?

The refs blow the whistle to stop play when a fight breaks out.

I don’t know a particular case but I’d be surprised if teammates never fought each other on the ice during a game because they do it anywhere else.

The first 5 seconds of this video is classic. The refs blow the whistle less than 2 seconds after the puck is dropped. But you can hear the whistle blow seconds after each fight starts.

Just like any other called penalty, referees blow their whistles and play stops, and the players watch as the referees do as well but step in once it appears the fight is won or both fighters are blown up.

The question makes me wonder what hockey would be like if they didn’t stop play when a fight broke out, and honestly, it might be kind of cool.