We had to go through this on and off with my SD. Seems like she'd go through phases with it.
We had better luck addressing the real issue rather than the lie. The lie is just because they didn't do what they were supposed to do and don't want to get in trouble. It actually takes more creativity to lie! We had to address why she didn't do the task in the first place.
My SD lied about washing her hands because she didn't like cold water. Instead of telling us this, she would just lie and tell us she DID wash her hands and then we'd yell at her for lying. Didn't work! Once we found out WHY she didn't wash her hands (cold water) we showed her how letting the water run until it was warm solved the problem.
My SD didn't do everything she was supposed to before bed because she wanted to hurry up and get ready so she could watch more TV before lights out. We solved this by shutting off the TV 30 minutes before bed, and the TV would not go on until the next day. She had no reason to go fast so she did everything on her list.
My SD lied about brushing her hair too. It hurt! A haircut, better shampoo and conditioner and detangle spray solved that issue.
My SD still lies about cleaning her room. She just gets overwhelmed, even at age 9. I give her one task at a time for her room e.g. pick up all your clothes. Now put away all your stuffed animals. etc.
My SD stopped lying about chores and tasks once we figured out all her resistance reasons. She will still say she cleaned her room when she really didn't do much, but I take that as less of a "lie" and more of a misunderstanding of how to clean a room since I get the impression that she truly thinks it's clean.
In each case you have to look at WHY your child is lying and address that. The resistance to do a task...why don't they want to do it? Sometimes it does just take a "we have to do that because that is the way it is," but bringing your child to that understanding will help a lot.
At age 5 she may be unsure how to do something, or need more supervision to stay on task. Good luck!