Fibrous rooted begonias (the small ones) are really just bedding plants. If you have space in a greenhouse you could try potting them up and trimming them back to give them a chance.

The larger tuberous begonias can be lifted and kept frost free over winter like dahlia tubers. Then you start them growing early next year - maybe March-ish. Watch out for the dreaded vine weevil grubs eating the tubers over winter. When I used to grow them, I'd check them every few weeks and dig out any soft spots - and the grubs - with the handle of a teaspoon. Squashing's too good for them!!