It's on our kitchen tap, which is only about two years old.
It's not this model but the key bit looks like this:
The little stick that screws into the barrel to turn it on and off has fallen out. On inspection the stick has a short screw thread just beyond the bit that is visible in the picture, about 5mm worth. The barrel is then hollow.
The problem appears to be that the metal of the barrel has eroded away to the extent that the stick bit no longer stays in. I thought of PTFE tape to make a better connection, but not sure it would be strong enough (and can't find it in the garage or I'd have just tried it).
I think it would be beyond my soldering skills to do that. Of course a different stick would do, doesn't have to be that one, but the chances of finding the exact size might not be too good.
Any better ideas?