"I had to wait for her to hang up. You hang up on a customer and you get fired."
Even when you're getting the blame for something that isn't possibly yours? They record the phone call, don't they? Challenge them by asking them to play it back. If they're going to fire you over what I term a "brick", then perhaps they're not the right company to work for in any event.
"She refused to accept that the physical line had to be repaired as I saw that she called in 3 more times that day. You get a little nasty note when you get a repeat call with in a week."
Again, point out she's blaming the company for something that isn't theirs. Doesn't the company have a policy for dealing with impossible demands?