The 9.77-tonne Dolphine went missing near the southwestern island of Chuja last night and its wreckage was found on an island nearby this morning, the coastguard said.
Three people were rescued and eight bodies have been found so far, it said, adding it was unclear how many were on board.
A public passenger record showed 22 people on board, but at least four people on the list were confirmed to have not boarded the vessel, it said.
One of the three survivors was also not on the list, it said, adding search efforts involving dozens of ships and divers were underway.
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"I was sleeping when the boat's engine went off and the captain told us to get out and water started to fill the boat," the 38-year-old survivor surnamed Park told Yonhap news agency.
Park said he and other survivors climbed on top of a floating piece of the boat and held on for more than 10 hours before being rescued.
A lack of safety measures and inaccurate passenger records prevalent among ship operators were heavily criticised after the Sewol ferry disaster that killed more than 300 people in 2014.