I had asked (had a flight leaving at 6pm) and was told no. I'm sure if you want to use your points up for an add'l day, you could do that. They'd always be more than happy to hold your luggage for you til later.
Due to the time required to prepare DVC villas, the policy is NOT to allow extended checkouts at DVC resorts. The resort will store your luggage until you are ready to leave and departing guests may use the resort facilities through their checkout day. If you need more time, you could reserve an extra day and leave early.
what's really cool is that you can do airline check in's at certain DVC resorts. When my sister and I were at the BCV in December, her airline participated, mine didn't. Right there at BCV, she checked in, got her boarding pass and didn't have to worry about her luggage or anything at the airport-our flights left at 6pm.
What a wonderful service!!!