As terrible as it may seem. You have got to nip that behavior in the butt!
You have got to let her know that she is not going to get her way if she does that! Act like you don't even hear her! You may even have to leave the room! Or tell her she has to go to her room or outside if she is going to act like that! Let her know that you no longer hear little girls who act that way! And let her know that when she calms down, you will be glad to talk to her! This may take a while, but she will come around if you stand your ground.
I personally know this from experience. I have a boy who is now 7, but used to be just like that. Someone gave me this same advice. At first, I thought it was cruel, but I was desperate because it was starting to get embarrassing.
After about 4 days of absolutely not giving in until he quit, my life finally started to get calmer. I was so happy! And you can be too!