No amount of asking, pleading, exceptional powers of persuasion, bribing, tricking, begging, threatening, shouting or cursing will get my DS1 (5) to do ANYTHING he doesn't want to do. Now, I ask 3 times to do something for example, put his coat on. If he doesn't then he'll just go out and be cold. He insists on learning the hard way.
Many times he has gone out without a coat / jumper as he refused to put one on.
Many a time he has lost / broken things when refusing to listen to advice.
Many a time he has gone hungry when he won't eat his dinner.
IMO the 3's are far worse than the 2's as they are learning that their actions can make them get what they want, they find it frustrating not being able to express their feelings properly and they have the word, "No", down to a T!
My advice would be to pick your battles carefully. It's not ideal that she won't put one on as it's freezing, but it won't kill her not to wear it and you'll save yourself sprouting a few extra grey hairs over the stress.
Not worth the argument. Take the coat with you, then if she wants it, she can have it.