I've seen many threads where ducks have dry peeling bills, leading me to believe that it is a thing that happens. Interestingly, I've noticed that most of the ducks are actually drakes. It could be a coincidence. My own ducks, all girls btw, have never had peeling bills.
I don't think the the peeling bills usually have those spots, though. My own duck did get some spots like that on the top of her bill. I took her to the vet. The vet said it was likely from her putting her bill through the wires of her aviary to reach bugs/grass on the other side and scraping it. She had me put a thin layer of antibiotic ointment without pain reliever on the bill daily, but seemed to feel pretty nonchalantly about the entire thing. I brought my duck back in for check ups and my vet was very pleased with how her bill was healing, even though it took months to heal entirely! So that makes me think that bills heal very slowly.
Of course your duck could be completely different, but maybe this will be helpful for you.
