You know how the old saying goes, "I learned long ago, never to wrestle with a pig. You get dirty, and besides, the pig likes it.". For whatever reason this lady is enjoying yanking your chain. It could be that she has an evil intent, or that she treats pretty much everyone rudely, or that she is mental, or all 3. Whatever.
If you get down and dirty with her you are going to get dirty, and she is going to like it.
And, like someone else said, she can only put you down if you let her.
Take the higher road, and kill her with kindness, or at least ignore her. You'll sleep better at night.