I think you'll find that the flowers and stems drop as the weather gets colder, but you can enjoy them until they do! I'm assuming that they grew from corms? Store the dry corms in dry compost or sawdust in a frost-free place and start them off again in the spring.

I bought them in as young plants and grew them on. They have already lost most of their tops from being frosted. I had expected there to be a good corm but having taken a couple up, there isn't much of one which was what made me wonder if they were worth keeping.
Do you still think it is worth a try?

No, Weedpatch. Sorry, but it does sound as though these have to be treated as annuals. The smaller the corms, the more risk of them shrivelling up while they're in storage.

Why not buy new larger corms in the spring, and then you'll be able to keep those from year to year? I do that - it does work.

There's nothing to lose in keeping whatever corms the plants have managed to make & seeing if they'll regrow next year. Just keep them dry and frost free over winter then pop them onto a seed tray of compost in spring and see what happens.

Thanks for the advice. I have been so pleased with these Begonias which I have never tried before, I plan to have more next year, so will do as you both suggest. Try keeping them but not be disappointed if they fail and get some better corms next year.