Unfortunately people like that usually don't realize how their words hurt both themselves and the person they talk about.

You shouldn't. If she were a close relative or a good friend of yours then what she said about you should matter, but she would tell you face to face. But she's not, and she criticized you for doing your job. This is very familiar to me... I've seen the same pattern many times: someone believes it's cool to criticize someone for their good job, something they should be doing in the first place but for some reason they don't and they'd rather make you feel bad about it instead of trying to improve themselves.

Don't let her interfere with your productivity in your job. It's hard to stay away from someone who works with you. If she says hello it's OK to greet back, but if she ever tries to pretend she's friendly with you it's up to you to believe her or not.

I'm someone who worries about others' opinions, too. And because of this I can tell you why you shouldn't. Personally, what I do is to compare how I and the other person are treated by others. Many students were nice to this classmate of mine and she always was the center of attention but almost nobody said anything good to her and if she needed something nobody cared, while I used to receive many compliments and I've been helped many times when I asked for help. Pay attention to the way you and she are treated by your coworkers. I'm sure you'll notice the same difference.