How do you deal with 6 year old bullies?

Today I took my girlfriend's 6 year old daughter (let's call her Iz) to the park, and after a while she came to me telling me that a girl hit her. I told her it was probably an accident and to keep playing.

Minutes later, she told me that the girl hit her again, and I asked if she was a baby or if she had done something that she didn't like. The answer for both questions was no. So I decided to put my book down and keep an eye out while she played.

She was waiting for the swing in the park to become available, along with other kids, and taking turns gently pushing the little girl who was swinging at the time. And then I saw it: A girl her size (skinnier though), pretty went running up to her and hit her twice in the face and chest, turned around and started playing again. Iz immediately started crying, and I went running to comfort her. I then confronted the little girl:

Me: Where's your mom?

Girl: I don't have a mom.

Me: Where's your dad?

Girl: I don't have a dad.

Me: Where's your yaya (nanny)?

Girl: I don't have a yaya.

Me: Then let's go. I'll take you to the police. You shouldn't be alone in a park.

Then another yaya that was there told me that her nanny was across the park, talking to other nannies.

I went up to her and told her that the little girl had hit Iz 3 times already, and that I expected her to be reprimanded, or grounded.

She said "ok", turned around and did nothing. I asked her to do it now, and I explained that kids should learn that hitting other people is not right, and that if she was incapable of grounding her, I'd be glad to talk to her mom or dad and explain the situation. She then went to the girl and took her somewhere else.

To calm Iz down, we went to grab some chocolate milk near the park, and when we returned, the little girl was still running around like nothing happened. Iz saw this and said that she didn't want to play anymore, obviously scared of being hit again.

So once again, I had to talk to the nanny and ask her what was going on. She basically wanted no hassle, and at some point she said that the little girl's mom had spanked her. To be honest, she was running around cheerfully as if it was her birthday that day, so I had doubts. In any case, she took the girl again and she told her to play on the other side of the park, so at least, after I went around with Iz to make sure that she wasn't there, she started playing again... until some boy started calling her "fat" and she got upset once again.

We decided to go to a different park.

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When we told her mother, she said that if someone hits her, first of all she should tell us; if the kid hits her again, then, push him/her as hard as you can. And if someone calls her fat, then say "thank you" and don't play with that kid. I guess that would work, but I wonder if there's something better to do given the situation.

So reddit, any ideas?

tl;dr: How do you not punch 6 year old bullies in the face and still get them to stop being assholes?