The cable attached to the car snapped, causing the deadly plunge in the Gais Valley, near the town of Chilas, Dawn newspaper reported.
Police quoted witnesses as saying the cable broke when the trolley was halfway across the river, apparently due to overloading of riders, the newspaper said.
A search and rescue operation was under way but none of the bodies had yet been retrieved as it was feared they might have been washed away by strong currents.
Works Minister Bashir Ahmed told reporters at least three area public works officials had been suspended for criminal negligence, Dawn said.
Other reports said five of the victims were from the same family.