Most Helpful Girl
Maybe she has something to prove to others, it sounds like she is keen but unsure of her own feelings and is trying to keep it at a professional level.
I don't think ignoring her will do you both any good but ask yourself the question, do you like this girl? Yes? Then just sit back watch and wait. If you don't then state your not interested. She will be weird at first but after her emotional whirlwind she will come to realize you're serious.
It sounds like flirtatious bikkering, sounds stupid but from what you said maybe she does like you and is putting her walls up. Girls are so complex lol.