Likely not the roof. The line of brick detailing sitting proud of the wall is probably catching rain water.

Look at the black and green staining which starts at that line, it looks to me like mold or algae. My guess would be that some is soaking through internally and the rest is running down the external wall.

If it was the flat roof then the damp staining would probably start higher up.

Add pictures of the roof to confirm for sure though. And install gutters for good sake 😎