I've been submerged in ice cold water before yeah. The pain is mostly in the bones I find. If you can get past that you can force your body to move, roll and hold your breath. It's cold though don't get me wrong and it fucking sucks once you're out too but it's not so cold that I can't control my body. Individual digits maybe but arms and legs no.

I fell through the ice once as a kid and I saved my cousin who fell through the ice, I partially fell in saving her too but managed to keep one foot on the ice so it was just my leg, I pulled her out by her hair cause it's all I could grab. She was able to walk home after being dunked in too again, in the middle of a Canadian winter. I've had way too much experience in cold waters, I also fished lobster for 2 years on a boat. Dunking your head does nothing but chill the ears, it's your body and your nuts that's the hardest :)