I can't wake up even with my baby crying

So I've recently become a dad, my little one is only a week old. The labour was pretty traumatic for both of us, so I'm hoping my problem is down to exhaustion.

Last night my partner said she would be fine up with the little one so I could get a few hours sleep, but needed a hand and I didn't wake up, even with my newborn making a fuss next to me with her.

Has anyone else had this issue and have any solutions?

I'm going to try to catch up on more sleep and also sleep when my newborn does to try to get used to waking up with him.

Any help or advice would be much appreciated.

Thanks