My daughter was the exact same way in first grade. She didn't pay attention, she didn't want to do her work (or refused to), would write answers that make no sense, etc. Otherwise she was a happy and social child.
I couldn't believe they passed her from first grade to second! We did some homeschooling over the summer to help her out and that's when I discovered just how deep it went. She couldn't read, she couldn't sound out words. She couldn't remember things, she couldn't understand things.
We ended up re-doing part of kindergarten and then we re-did all of first grade. Then we decided to homeschool because she did the best with one-on-one instruction. She doesn't have ADHD. She did have some issues following directions "because she didn't want to" so basically she was a willful child but we brought all that under control.
She would be in 5th grade this year, but she's doing 4th grade curriculum. She's advanced at one grade per year if you count having to re-do first grade. If she was in public school, she would have benefitted from being held back.
What I have discovered homeschooling her is that sometimes their brains just aren't mature to "get" the material when it's taught. For example, I was teaching her how to read an analog clock and it was taking forever. She was frustrated, I was frustrated so I dropped it for a few months. 2 months later I picked it up again and she got it no problem! I didn't change what I was doing or anything. I just gave it some time.
So if homeschooling isn't your thing (and it's not everyone's thing) then I would suggest pursuing holding her back a year. Based on her personality and and such, it could be the best things for her.
Kids who are held back may drop out of school later on because the root of their problems were never solved. Try to work with someone to find the root of her issue. For my daughter it was reading, she needed a lot of help reading.
Good luck!