Yes, there is.
Each park has a First Aid Station. They are not just for first aid, but can also be used by people who need a rest. We've taken our DD to lie down there when she needed a rest in the afternoon. They have cots in individual small cubicles, divided by privacy curtains. It's cool, dim and quiet there and the staff arre very nice.
If you are saying on property, especially at one of the deluxe hotels, it might be worth considering going back to the hotel for a mid-day break. This is especially true if at the MK and staying at one of the monorail hotels, or from Epcot or MGM to either Y&B, Boardwalk or Swolphin.
If an adult needs to lie down in First Aid, would another family member need to be present? I mean, would they just let you drop her off and then leave?
In my DD's case, she can't be left alone, so I did have to stay with her. But, if the person who wants to lie down is old enough (no small children) and competent to be there by themselves, I don't see any reason they would need someone to stay with them.
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