Are there any products that are effective for exfoliating the arms and legs?

Now that the weather has turned chilly

Even if you moisturize your arms and legs after showering

I could see it turning white...

At first I thought it was dead skin, but seeing as it kept peeling off even after I scrubbed it, it seemed like dead skin cells.

I can't scrub myself often because it hurts too much.

It seems like I'm getting more dead skin cells...

There are various peeling gels available for facial exfoliation, but I haven't tried them for the body, so I'm not sure. Could you please recommend a product that effectively exfoliates without irritating the skin?