Why can't it be "No, I'm no student"?
That's not a very common way to say it, but it's grammatically correct, so I think it should be accepted.
I'm not a student suggests that you do something else.
I'm no student is a self deprecating way to suggest that you find studying hard.
Oh, right, I see! If you want an equivalent of that feeling in Ukrainian, you will either just use intonations, or maybe say "З мене ніякий студент" ("I make no student") or "З мене поганий студент" ("I make a bad student")