Yes, but it would probably require flippers, being a lot lighter, having more strength, and extraordinary weather conditions.
The amount of upward force produced if one assumes that the pressure on the "wing", is proportional to the surface area of the "wing" that faces downward. Wearing giant flippers would definitely help because that means more contact with rain.
In order to fly with these gigantic (and presumably heavy) flippers, one would need a lot of strength. This would allow the user to exert more force in total on the flippers, and would also allow him/her to "swim" faster.
I suppose being a lot lighter isn't really a requirement, but would help make flying easier. After all, that weight is dragging you down.
Finally, we have the weather conditions. The largest raindrops ever observed were about 8.8 millimeters across. Laboratory studies show that theoretically, the diameters of these raindrops could approach 1 centimeter (if isolated). Ignoring the isolation rule, one could, using super strength and giant flippers, probably swim around with much difficulty. In order to ease the workload on our pour user, it would be much more helpful if the rain came down in sheets like @Aethernolt said.
TL;DR: Your guy can swim if he has giant flippers, a lot of strength, being light, and being in the middle of a tremendous rainstorm with giant raindrops.
Troll Method: Make the guy lighter than air.
Edit: Lighter gravity, which causes less downward pull and allows for the formation of larger raindrops, would also help. In fact, if the atmosphere was thick enough (with water vapor from the rainfall, for example), and the planet's gravity was sufficiently low, you wouldn't need raindrops. You could just strap on some wings and fly. (If this method works as well, see https://what-if.xkcd.com/30/)