Given the location, around the base of the faucet and under the handle, it looks like hard water deposits to me. My mom has dark faucets and granite and a white film develops where water comes in contact often. Maybe look for something like CLR and test an area first.

Thank you! Someone else suggested CLR on another one of my posts for stains on my faucet!

Yes, I can see the same hard water deposits on your faucet to. We have hard water, and even with a softener, we battle deposits.

You can try topical treatments, but pulling the faucets and getting a skilled operator with some diamond abrasives will probably get rid of those. Keep the water off religiously or you're having him back.