Is it OK for kids to ice bath?

Children typically want to imitate their parents, so when they see Mom or Dad getting in the ice bath, they want to try it, too.

Is the cold water safe for such small bodies?

It sure is -- as long as they're supervised by a parent.

Children have two big advantages in the cold:
1) They have more brown fat than adults, so their little bodies can generate a lot of heat, and
2) They have the good sense to get out of the cold when they're not OK with it.

Never coerce, bully, or shame anyone (especially children) into the ice bath. When children are free to explore the cold water of their own volition, it's safe for them to challenge themselves just like you do.