Hi. Sorry no one has responded to your post yet! I did not see anything in my search that says that banana peppers are toxic. But, I am surprised your cat wants to eat any of the parts! If your cat has eaten them, and it is harmful to them in particular, you would know - vomiting, drooling, lethargy, diarrhea, etc. But, the bottom line is: keep your cat away from them, if no reason other than, there is no nutritional benefit to having a cat eat them.
OK...that sent me on a mad scramble of searches. NOTHING out there mentions banana peppers. HOWEVER...I was able to find that BELL peppers are not toxic to cats (although in large amounts they can cause tummy upsets), AND that banana peppers and bell peppers are very closely related. Hope that helps!
If unsure, don't give any to the cat. If the cat ingested some, keep an eye on the cat for the next few days and watch for any signs of illness and be ready to take the cat to the vet if needed. This goes for anything edible and non-edible that you are not 100% sure about and the various lists of toxic food and stuff for pets online have no info.
Keep the free poison control hotline number in an easy to access spot or in your phone just in case you need to call:
Kansas State University Veterinary Teaching Hospital 1-785-532-5679