Hi The Fox & The Hound, sounds like she has some things she's found/finding it hard to deal with. Maybe things she hasn't told you about yet or things she has put to one side (maybe not even dealt with yet?).
If she doesn't like being by herself even for a few hours and is also saying she doesn't like being herself then, yes, maybe something wrong.
What sort of a friendship do you have, do you talk about more personal things? because maybe she's not talking more about what's going on for her if she senses a more "casual" connection on your side. I'm not saying there's anything wrong with a "casual" connection, but................
Or maybe she just wants company, "no strings attached", "normality", and to put whatever is going on, on the "backburner" while she has this.
But whatever is going on if you want to stay "friends" your needs matter AS WELL, so if you want more space you really need to tell her that (inc. give her some examples), and as a friend she should respect that. Maybe you'll have to compromise just a little, but there do need to be some boundaries there as you're obviously feeling uncomfortable with it all.
And you can "break it gently" to her e.g. tell her that you really want to be friends, you do really like her (if it's true) but.................
Then I don't know if you want to come back and talk to her (letting her talk too!) about what's going on for her on a more personal/deeper level. I mean she might not say anything anyway, but she might, or if not then maybe later or if your friendship develops.
The balls in your court on that one, depending on how you feel personally
But do you think you might help as well to go out together with other friends too, maybe build up her social circle as well, so it isn't as much all about you.
Alison