I am working in a private bank as an executive and now, I am planning to do MBA in banking and finance. So, am I eligible?

Any graduate is eligible for MBA, without any doubt. You can also give many entrance exams like CAT, etc. If you are planning to study while working, then I would suggest pursuing Executive PG Diploma in Banking and Finance. Since you already have some experience, it would be good for you.

Yes. You are eligible for MBA in banking and finance. Do share academic profile as it plays an important role at the time of admission process (after CAT/XAT and other tests).