You may have a trailing form of African violent (Saintpaulia) - these have a tendency to form baby plants on the tips of the drooping parts, and will continue to do so even if you remove the existing ones. I can find no information as to whether these will develop roots if pushed into compost, the usual method of propagation for these is leaf cuttings. I suppose you could try and see what happens.

Thanks Bamboo. I had no idea there was a trailing one, maybe I will just try one and repot the plant.

I've just noticed I typed 'violent' instead of violet in my earlier answer... not a freudian slip so far as I'm aware, lol!

Put them on the birth pill

Mine once produced a baby at the side and I potted it up successfully.