We perceive water as a fire extinguisher, based on observations related to the atmospheric conditions on our planet. However, Let's see if we can circumvent this limitation. Actually, water extinguishes fire because it forms a separation between the fuel (such as charcoal) and the air around it. That's why we can't extinguish gasoline fire with water, because the gasoline floats on water and remains in contact with the air.
Assumption: Carbon dioxide, dissolves readily in water. On the other hand, oxygen is less soluble. Now, let's say that the atmosphere of some planet was very dense (something like Venus's atmosphere or more) and yet very rich in oxygen both in percentage and pressure. Pressurized oxygen at sea level means there will be lots of oxygen dissolved in the water. Carbon dioxide is soluble enough to be moved away from the fire location, so in theory water will be unable to separate the fuel from its oxygen supply, and therefore even "wet" objects may catch fire.
Some points to consider: The fire should neither occur from chemicals reacting with water, nor chemical mixes capable of combusting without additional oxygen. It must be something that (For instance) a primitive underwater society should be able to maintain without complex chemistry and physics.
Clarifications: Combustion should be maintained via reaction with oxygen dissolved in water, whether naturally or not. I'm not speaking about electrolysis to obtain oxygen and hydrogen like this question here. In addition, I'm aware about the heat properties of water. i.e., water absorbs heat more readily than air, so that the fuel is cooled down and it is therefore much harder to maintain fire underwater. You can see this video, where a firework stick extinguishes in water very easily even though it does not rely on oxygen from the air. On the other hand, the same stick isolated with tape can burn underwater without any oxygen supply. You may cause combustion by heating the fuel. One way is to expose it to hydrothermal vents. One other way is to supply heat from other sources, such as a laser beam which serves to maintain high heat at a pinpoint location.