Yes mosquitoes can bite you when you're sleeping as it's happened to me.
Of course the mosquito biting me woke me up but some people say they've been bitten by mosquitoes and didn't know it until the next day when they woke up and saw the bite.
I sleep well protected under covers so that mosquitoes won't be able to get to me while I'm sleeping.
Some mosquitoes are more active during the daytime while other mosquitoes are active more at night.
I would recommend also spraying yourself with some OFF bug spray to keep the mosquitoes away from you both during the day and during the night before you go to bed.
Or wear a clove of garlic around your neck while sleeping as the mosquitoes hate the scent of garlic.