That’s the expected burn. What surprised me, was that the non-sensitive area "hand" would burn, and that it would seep below the top layer of the skin so you can’t even clean it off.
Rubbing your eye after handling peppers is a definitely a problem - I’ve had that issue when peeling roasted green chile. Going to the bathroom can be risky, too. I had a vivid experience with that after chopping up a couple jalapeños at Taco Cabana one time…
Avoid touching sensitive areas if you handle chili peppers or hot sauce, eyes, nose, groin...