This bug occurred to me twice.
The first time my character would not heal(even after days) after fainting in a battle.
I them started a new game.
This happened to me again but I was able to find out why.
When i was knocked out this time I noticed that my character had "Died or had been fatally wounded" in the battle summary.
In my party screen I noticed Qyburn became the top of the board while my name was missing. Qyburn could not be moved up or down and I could not talk to him.
Would turning off perma-death for companions fix this glitch?
Companions don't have perma-death. That is part of the bug which has been present since Brytenwalda. The companions will come back eventually though..
To get companions back it helps to send another companion on some mission, 'connections' , 'support your claim', usually the 'dead' companion will come back before the one you send out does. I've used it in Brytenwalda many times and a couple times in AWoIaF.