Baby wakes at 4am to play

My baby (almost 11 months) has woken up at 4am the last two days wanting to play. She’ll wake up at 1am and I get her back to sleep and in her crib but then when she wakes at 4 she’s wanting to get up and play but is still super tired and then throws a temper fit because she’s tired. I try to keep her in the dark bedroom, no talking to her, and I’ve tried nursing her to sleep and she gets distracted and doesn’t want to at first. Any helpful tips?!