Out of all the women on earth,

[beaubrummels]

she maintained that she hasn't decided not to go out with me but she needs to know me better first. However, every time I try and make an effort she seems to withdraw.

I think you need to double date her, or only meet up in public places, with friends, etc. It sounds like she doesn't know you well and doesn't want to be alone with you yet. You have to think of her view point. As far as she knows, you could turn out to be a serial something-or-other. Until she is confident you are a decent man, you are going to stay outside the circle.