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That's a good point to check.
The run starts from the other side, but I'm not 100% sure on the exit. I know that ceiling leaks can be tricky sometimes as they don't always correspond with a roof leak.
The more I think about it, the more I think it might be condensation build-up under the paint.
The thing I'm unsure of is what to do after to dry it / strip it. And what paint to use to protect it.
It's a pain as we'd ear marked time to do "fun" redecorating DIY... not proper home maintenance.
A bit of speculation: is your damp patch close to the lowest point of the fan ducting?
A couple of winters ago I had a sudden brown water issue (doesn't sound right however I phrase it) staining the ceiling and dripping from the fan switch: water in the duct had pooled and then found a way out.
My best guess at the time was the cold duct wall (unheated roof space) was allowing moisture to condense before it reached the outside - a shorter duct and lagging (a roll of garden fleece) seems to have solved the problem so far...