I don't know how it is with her, however, this woman after finding out from our mutual friend that I was going to ctb, started threatening me that if I do it she will too.
Her life, her choice.Not knowing the parties involved, I'd suggest this smells of drama more than it does of serious suicidality, but I certainly could be wrong.
FWIW, I think I'd be at least as pissed at the "mutual friend" who is blabbing your secrets as I would at Ms. Drama Llama here.
At this point since at least two people you know who are incapable of keeping secrets -- so it seems to me-- know what you're planning, I'd suggest a "wellness check" from the police to be as likely as anything else. And if you play that wrong an involuntary commitment as a follow up.
Live your life as you see fit, for as long as you feel like living it. Don't give in to emotional blackmail, either to stay alive or to end your life.
I'd also strongly suggest not being public with your plans to IRL friends and family going forward, but that is obviously your decision and not mine. My personal experience in this area is you can't really trust people when the topic is suicide. And you've already had your trust betrayed at least once, it sounds like, unless you gave the "mutual friend" carte blanche to discuss this topic with others.
On the one hand I don't want anything to happen to her,
That's a perfectly fine sentiment, but at the end of the day you are you and she is she.
Honestly, and as others have noted, it sure sounds like you're being manipulated here, but again all anyone here can see is what you're telling us, so you might be unintentionally leaving something out or we're all drawing the wrong conclusion.
on the other hand I don't have the strength to put others over my own feelings again. What should I do?
I'm kinda getting a sense that you're having doubts about the whole killing yourself thing, apologies if I'm overstepping some boundary here. I mean, you ask twice "What should I do?" in that brief post alone. I'm absolutely for free choice in matters related to ending your life, but I do think anyone going down this road needs to make sure it is the proper course before proceeding. And I'm thinking you've got doubts, beyond this situation. A matter of the tone of the post as much as it is anything else, I guess. You just don't sound like somebody who's made up their mind. If I'm wrong, feel free to tell me to fuck off. Wouldn't be the first time.
To reiterate: If you decide to CTB do it for yourself. If you decide to live, do that for yourself as well. Don't be blackmailed or bullied either way.