Sarah's answer is correct. First you cannot "extend" a package at a resort. Packages, including dining, are only sold by Disney Reservation Center. A resort cannot sell them. And Dining packages (paid, not free) must be set up at least two days prior to arrival of the guest at the resort.
So if you are at WDW on a package and decide to stay longer, the resort can make you a new reservation for one extra night. But it is not part of the package. Like Sarah said, the Dining portion of a package is good until midnight of the day you check out (last day of package). So you can use any left over components on the day of the new reservation.