It looks bad actually, will provide pics, is this a common problem?
Is it as simple as going to a body shop and have them repainted or get new mirrors assemblies?

I dont see any rust. I see paint peeling and oxidation but no rust. It's probably due to salt where you live. The base is fine, Just have them redone

I was in the same situation as you, I sanded the base, where it was peeling off ,after I've putted primer and painted with a matte satin black spray, that's it !

I would bring them to a body shop to have them painted properly.
Recently done mine and they did a perfect job, can't even tell they were painted.
I suppose its due to the fact satin black is easy to paint :)
They look a bit glossy in the photos, but thats just the iPhone camera.

Did you remove them from the car on did it while attached to the car?

You'll get a better finish taking them off to paint, especially if the corrosion wraps into an inaccessible area.