Why would a girl constantly go out (alone) with me, and get upset when I talk with other girls, but not want to date me?

Anonymous
I've asked this girl out 3 times. She has rejected me 3 times, but for some reason she keeps texting me to hang out. We do things like go out to dinner, go to the movies, play mini golf, go stargazing, etc. all these things that are normally dates but she insists that we are just friends.

When we are together she does things that she doesn't do to other guys. She'll hug me randomly, she likes to ask me about my friends and family; when there are other girls around that are dressed a little "exposed" she will say things like "Oh don't look at that girl". I am pretty much the only guy she hangs out with, she'll talk to some others but never go out with them. When I talk to other girls she doesn't care but she always has to ask about who she is and what my relationship to her is (she doesn't do it when I talk to guys though).

I am fine with just being friends with her, but the issue is other girls see me with her and get turned off because they think we are a package. I've tried to tell her that I can't keep going out with her, but she still wants to. It's like she wants to date me, but everytime I bring it up she denies liking me as more than a friend.
Why would a girl constantly go out (alone) with me, and get upset when I talk with other girls, but not want to date me?
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