Summarizing Important things to do:
#1 - Don't push her at all, if anything, completely ignore the situation - for the time being anyway. Before you do this, a small, short, simple comment like 'I'll let you talk about it when your ready' might help.
#2 - When she is ready, don't be pushy about anything. If she really, extremely starts to open up, then TINY, TINY bits of asking questions MIGHT be acceptable.
#3 - Be careful how you word things around an avoidant. They will read into it more than you can imagine. The more 'carefree' you seem around them, the more they are likely to relax around you.
As an avoidant myself, I can tell you the people who raise my anxiety the most are those who I cannot read, or seem to observe alot more than I would like them to. If I can't get a general idea of what they are getting out of a conversation or situation, I will start to think they are getting more than I want of them to get, and will stop it there.
You'll notice most of my points involve not pushing her about it. The second she feels like your pushing in on her, or digging into her, or trying to know more about her than she feels comfortable about, is the second you will get shut out and pushed back to the beginning again. Take her out of the spotlight. This will, after some time has passed, allow her to take down her defenses some.
AvPD is something that, for the most part, 'active' pushing is not likely to fix. I'm sure this has got to be extremely hard on a worried parent, but your best option, atleast with how she is reacting now, is to leave her alone for a while.
However, make sure you keep general conversation up. Talk about whatever is of interest or importance. Small conversations about whatever, not related to herself. Even if its a simple talk about a TV show you like, or whatever.
From my experience as a AvPD, I am more likely to open up to someone who is more of a 'friend' than a 'parent' figure. I'm not saying drop the roll as parent - but about whatever situations you can, just try to be more relaxed about it all.
And a few quotes from others i'd like to emphasize
Dont push her into socialising or getting a boyfriend, or else she will hate you for the pressure you are putting on her.
Stuff like this feels 100000000 times more pressure than you think it is for an AvPD.
Lastly:
Now she is imagining that even I am saying hurtful things that no one else would see as hurtful.
As an avoidant, I know its very easy for me to take the simplist of comments, and twist them around to mean something that they obviously don't. That doesn't make me feel better, nor does it ever remove the belief that they are true. I will then also search for evidence that those are true, and can just as easily twist something else around to be that evidence.