At 9 months old she probably needs two naps. So she is thinking bed time is a nap so then she’s up again.
My LO didn't drop the 2nd nap until closer to 18 months. Your baby definitely needs a 2nd nap, as you said, an over tired baby is harder to put down x
At first I was going to ask how old baby is but everyone is saying 9 months, if that's right I'd def try the 2 naps. My daughter didn't consolidate to one nap until almost 18 months. If she's up at 7 try for one at 11 and another at 3. Just make sure she "plays" hard before both for a few days and set the routine
Maybe a bath before bed time and some sleep sounds , essential oils
my son is 9 months & he gets 2 naps & his bedtime routine is strictly only done at bed time that’s how he knows the difference from a nap & bed time. nap time we only change his diaper & offer a bottle . bedtime is a whole routine! yoy got this momma try 2 naps my little one sleeps better at night with 2 naps vs if he only has one he doesn’t sleep as well!
Look into self soothing. She may not be able to if you’re immediately picking her up. I’m not suggesting to let her cry it out. Pay attention to her cues