I had selective mutism as a child. Would only speak at home, never at school (at least when I was very young).
The most important thing you could do for this girl would be to not in any way indicate that she is inferior for not speaking. That can cut to the bone and can be very easy to do unintentionally. For instance, once when I was in high school I happened to overhear classmates whispering about how I "never said a word." Although they weren't being catty, they weren't being complimentary either, and it hurt to know that my silence was noticed in a negative way. And, hell, it's been since the beginning of time that I was in high school, and I remember that like it was yesterday.
Don't make a big deal out of the fact that she doesn't talk. If you make great efforts to draw her out, especially in public, she'll get the idea that she's defective. At least I would. Just treat her like a friend. Don't force anything. It's probably not your job to treat her condition anyway. Enjoy her for who she is, and who knows, maybe one day she will open up to you.
If she does, you may want to get some idea why she is selectively mute. Such information may help you approach her in a more helpful, non-threatening way. In my case, in addition to AS tendencies, I have expressive dysphasia, which means I have trouble with finding words to express my thoughts (I often do not think in words). Also, especially when tired or anxious, my speech can become garbled. All this makes me anxious in social situations, hence the mutism. If your friend has similar issues, you could give her a longer time to respond--ie, don't become impatient with her as she struggles to find the right word. Just an example, of course, since I don't know her.
Best of luck with this and good on you for wanting to help this girl.