What kind of thing would you say that she gets offended about? How old are the two of you? I've got to say it all sounds a bit petty. Word of warning - if she's been getting offended by the things you say, if it comes down to which of you keeps your job it will be her.
My advice would be not to ignore her (that's extremely rude - however uncomfortable she's making you feel, imagine how much more so you are making her feel). Instead, converse about topics that are not important and unlikely to cause you to say something that upsets her. What is it she has been getting upset about? Is it when you've given your opinion on things? So she's not the type that enjoys a debate. Don't discuss controversial topics and keep your opinion buttoned down. That's quite common and will be the case with many people wherever you work.
With pretty much every job you ever have you will find there are people there that don't mach your personality. At work we sometimes have to be a chameleon and behave differently depending on who we talk to but without being rude (which ignoring most definitely is). You need to get used to dealing with it. Honestly I think speaking to your boss about it is a bad idea. It will reflect more poorly on you than her. Also, if one of you ends up losing your jobs over this will that make you feel good? You can't get along with everyone, but part of working with different people is that we have to be capable of doing our job professionally regardless.
I dont say offensive things, shes just really sensitive, and you're on the mark when saying she doesnt enjoy a debate, she gets mad when that happens too. By things that I say, it could be something petty, the most recent example is I noticed she had a new dress, and said it was "bright" I honestly dont know if I like it or not so I wasnt going to say either way. She said something like so you dont like it then, and I thought about the question to give an honest response. Before I could even say something (I was going to say I liked it as a skirt) she had called me a profanity.
I've been working here for a few years now and have seen and worked with many many people. I've not liked some of them but I could easily work with them fine, but working with her is just on a whole nother level. I think I'll see how the next shift with her goes, if its bad, then I'll go to the manager. I'll try my best to have it not bother me, but I dont know how much control over that I have.
@Terra Nova It does seem that way, when she asks something she expects a response she wants to hear. I think you're definitly right that there can really only be a professional or extremely fake "friendship" relation between us.