Just when you thought you had it all figured out, I have to warn you of a New Madness that lurks just under the surface of western soceity. Some people peel things. Same goes for oranges. But other things not only are totally without need of the attentions of your peeler, but are even done a disservice by it's use. The fact is, there's just no need to do it.
Mushrooms grow from the soil.
This can easily be wiped off with a piece of kitchen roll prior to use. They don't even have a skin as such. May as well go out and peel a strawberry. Leave the skin on.