Silene aegyptiaca is my guess, but Desmond Meikle (Flora of Cyprus) does not have it in this area of Cyprus.
It was plentiful in a particular damp vineyard between Peyia and Kathikas (Paphos district), but I had not noticed it elsewhere. So perhaps it was imported from elsewhere on the island in a potted vine, and that might explain why Meikle does not have it in the area. (The damp vineyard is a typical habitat, according to Meikle; and the flowering time of March is right.)
The photo
here certainly has a similar jizz.
But the petals on my plants all have a tooth on each edge.
This photo and description include those petal teeth.
So, like I say, S. aegyptiaca is my best guess.
If anyone can confirm or correct that, I'd appreciate it.
(Apologies for the photos being a bit poor, but it was dull and windy.)