@AutumnalLight Late to this but wanted to weigh in. My mother killed herself just over a year ago.
We haven't really told many people. We just don't want to. If we see people who don't know, we say 'she wasn't very well'. I am extremely choosy about who I tell, because I don't believe many people can handle it without being insensitive. (However, you sound very gentle and sensitive.)
Your friend is probably still deep in shock. For me, it took around six months to wear off. I'd say it hasn't fully worn off even now. Suicide bereavement shock is something else. Her brain is working so hard to protect her, and it's only letting her process a tiny bit at a time. The idea of talking to you about it – no, that's clearly too much for her right now.
I recommend just keeping on being there, gently and subtly. Check in regularly and lightly. Share things with your friend – the stuff you used to share, whether it was recipes or memes or whatever. Keep being her link to 'normal', be a rock. If it's her cup of tea, have walks together. Just stay the course, like the real friend you are. I think she will talk to you, one day.
Buy her this book – it's often mentioned by people bereaved in this way.
Be aware that the inquest will be coming up at some point in the next 6-8 months (I doubt it has happened yet, the delays are crazy). I wouldn't mention this to her right now but do be quietly aware of it, because that will be hard for her.