This will probably sound inconsiderate, but it's impossible to "judge" the scenario here without hearing more than just your side. I'm only mentioning this disclaimer to indicate that in the following my intention is not to judge (even tho it looks like it contains judgements), it's to simply give you perspective and ideas / strategy thinking about the situation and what to do. Second disclaimer: also, without really looking deeply and clearly at the situation I can't have a great read on it, so liberally intersperse the following with plenty of modal language like "probably", rather than the authoritative unequivocal tone in which it is delivered for brevity and clarity. Third disclaimer: I'm not giving you advice or telling you what to do, there's no karmic connection between us and I have no responsibility for what you do, what you do is up to you, and how you interpret what I say is up to you. If you take this info in with those disclaimers, you are welcome to it! Otherwise you cannot have it! :P

"She has many friends in the department I have none". This to me is the clearest indicator of what is going on: She's just exercising her social dominance; This is not about you; She has decided she doesn't like you^0, and there's probably not much you can do about it, because: even if you had the means to communicate clearly to resolve any misunderstanding, she may be set on framing you as at-fault, so she won't care. And second, if she is just flexing her social muscles and power here, she will easily turn all her friends against you, and will certainly rely on rumours and accusations. She just wants a scalp, to show her power, that she can execute someone if she wants to. This gives her a sense of power and control, which is really important for her to feel at ease. So you were, by virtue of your lack of social capital, simply the low hanging fruit. She will leave a trail of such scalps behind her as she attempts to express her social power to bring herself more power and control.

Also if this is the situation, I don't know what you specifically can do--I know I could do some things, but I don't know you. So let me ask you: if this is the situation (for the sake of experiment, assume it is) what is your plan now?

Maybe advice from people who know how to turn a social disadvantage around to their advantage would help, but maybe you need to just take the loss and the lesson here (the lesson is not avoid pissing off toxic people, it's, I think: cultivate relationships to ensure you don't lack social power and make yourself a target) and move on to greater things.

^1: You may have done something to irk her or maybe not, I don't know. But even if something you did pissed her off, you didn't do something wrong objectively. It may have been her wilful misinterpretation of something you did, in other words, she seeks fault in you simply to create a fake justification for her pre-existing dislike of you that that she doesn't understand, and then can use as a way to "lift" her personal dislike to the level of social censure with a fake story of your badness; or maybe she genuinely thought you were shading her, when actually it wasn't like that at all for you; or maybe she is simply a predator picking on you because you are weak and she is strong (socially). It's not that she dislikes you, it's simply that she needs a scalp, and you are the low hanging fruit.