Do people have ongoing tenancies (no fixed term specified)?

The lease on my flat ends in a couple of months and the agency has sent the standard email asking whether I want to extend for another year at a higher price etc. Anyway, the year-long tenancy contract is a nuisance because I want to move in around about 8 months, but the precise date depends on some factors which have yet to play out.

Could I say to the landlord that I'll stay on in the flat indefinitely, and then give like one month notice or so when I want to move out? Do people do that, or is that not really a `thing` in the UK?