I doubt it. I thought they were immortal.
Tell that to the goldfish I had when I was a kid...

This is just pure speculation on my part but, I am pretty sure pets are supposed to be able to assist a person in fighting so, I would think that there'd have to be someway that they can be knocked out but, would doubt they would die... maybe just get hurt and have to return to your magic ring until they can be revived at some later time? *shrugs*
Anyway, pets are seriously bugged right now and can't even stay "summoned" for longer than a few minutes so, while a good question, not one you really have to worry too much about right now.