Okay, my daughter is 8 and in 3rd grade. As are about 3/4's of her class.
When my daughter is in 4th grade, she will turn 9 years old.
No, she did not skip a grade.
My daughter will be 9 in 4th grade.
My daughter is mature and just fine socially. And on par. And has no social or maturity problems.
I don't know why, it is so 'bad' that she is going to be 9 and in 4th grade. As I said, many of her classmates, are also her same age.
I was that age, when I was in 4th grade. I was fine, too.
I graduated High School at 17.
Now, you cannot turn back the clock, on how old your daughter is and in 4th grade.
Besides, holding her back.
But that is not an option for you.
Your child's 'problem' as you said, is her maturity.
How is she at home?
Does everyone treat her as a 'baby'???
Or do they treat her her age???
If she is having a hard time adjusting, socially, then work with her this summer.
Enroll her in activities/clubs etc., so that she gets more exposure to other kids and ages.
Talk to her.
Ask her 'why' she wants to act like a baby?
Maybe growing up scares her?
There is a GOOD book series called the "American Girl Books." Which is for this age. It is child appropriate, easy to read, and meant for the girl and Mom to read together too.
Just look for it at bookstores. They have it. Or online.
You need to talk with her.... with her. Not at her or down to her. But talk with her... about her feelings, why and how come's.
I don't know that you 'can' change a person's 'maturity.'
Or, talk to counselors or your Ped.