Is this a normal reaction to a "superficial" chemical peel? My face is still burning and stinging. (photos)

Though it is too early to tell, the splotchy areas in my experience are uncommon. You did not say what type of chemical peel you had (there are many).
Best advice would be to protect the skin. You should have been given a skin protective product of some sort to apply that will protect the skin.
I would return to the physician that performed the peel and let them know your concerns.
Good luck.

I recommend following up with a cosmetic dermatologist for recommendations after a chemical peel including silicone gel with growth factors, lasers, microneedling/prp and skin care. See an expert. Best, Dr. Emer