Basically yu have to open the ceiling as much as possible and ventilate it to dry it up. Damp isn't much better than dripping wet as far as future rot & mold are concerned, so get it dried somehow. If there are lights in the ceiling, remove them and rig some sort of tubing to duct forced air in there. Better yet, drop the ceiling upholstery altogether, if you can figure out how it was done in the first place.