This is a common problem and is caused by a failed tailgate lock actuator. Usually when they fail you can manually lock the tailgate from inside the tailgate and the key/remote will unlock it. However, due to the push/pull mechanism breaking you cannot lock it with the key/remote. Time to replace the lock actuator mechanism inside the tailgate.

A search of this forum will provide instructions on how to do this if you are so inclined to repair yourself.