I dont know.
If a friend if mine said "I know what you're like" then I take it to mean they know what I'm like. They've known me for years, they know me, they know how I react to things and they know how to sit with me and talk if im upset. Just like I do for them.
You've taken it to mean "I know what you're like" as in, you're ridiculous, silly, over react and she is used to it so can ignore you, cope with it and doesn't mind if you do that.
She was still wrong. When you told her you would find it too difficult and she said she would be there for you when you're feeling like that because she knows what you're like, she was still completely wrong. Because the reason you'd be feeling overwhelmed would be her being there, so sitting with you when you're upset wouldn't help. But she could have meant it like that; that she'd be happy to help.
She was wrong. She wouldn't have helped. She should have listened to the no the first time.
But I wouldn't throw away a friendship over this without talking fully about it first. I wouldn't back down, I wouldn't apologise and I wouldn't be gentle gentle with them. But I would give them the chance to talk to me and let them apologise if that's were they were going. If not, then I would take your route of not speaking anymore.