I have a three year old too. I understand you are frustrated.
but you’re turning everything into a battle. You’re not going to break her and she’ll finally start complying. That’s old school discipline and it just does not work on some children. Yes, it will work on others- they’ll be afraid of there parents and just do as their told, but some just won’t and you’ll never ‘ break her ‘.
my son is like your DD and my older DD who’s now 5, was also very similar. Tantrums ALL the time. Refusing to get dressed, running off, refusing to brush teeth- running away, not walking to the car, constant tantrums. BOTH my kids are / were like this.
hard arse discipline doesn’t work with these kids. My brother was like this too. My parents shouted, hit etc. it only made his behaviour worse.
you know what does work for kids like this ? TIME and patience and understanding.
Rewarding good behaviour, rather than focusing on the bad all the time. Ignoring bad behaviour sometimes and just distracting them. Stoping the power struggle. Preempting when the power struggle will start. Preempting bad behaviour - giving them choices.
for example if she doesn’t want to walk back to the car after nursery - you try telling her that if she walks back nicely - she can have a start or a treat or whatever it is. Or you race her back to the car / or you bring her scooter so she can scoot, anything to stop the current routine and power struggle that ensues.
getting dressed in the morning - make it a reward or a fun activity - give her choices of outfits - or order of how she will get dressed - shoes first or jumper first ?
I know I don’t have all the answers here, but you cannot continue to battle every little thing out with her in my opinion.
it’s a stage. She will change and she will grow out of it.
they also do it for attention- yes, even negative attention, is still attention.
I speak from experience. My DD is 5 now and she’s pretty well behaved and she was an absolute nightmare at 3 ( like my son is now). They grow out of it as they mature. It doesn’t mean she’s a bad kid or will grow up to be a bad kid. I bet if you weren’t so combative, she’d stop hitting etc.
good luck