Yes, in most of Catford, and plenty of people already do. There are few listed buildings and only one conservation area. The neighbours file planning applications all the time. The real constraint is underground. The whole district sits on clay that shrinks and swells, so any extension needs foundations designed for it.
Historic England lists 30 buildings in Catford. None is Grade I, one is Grade II* and the other 29 are Grade II. Listing covers the inside as well as the outside, so any change that affects a listed building's character needs listed building consent on top of planning permission, and doing the work without it is a criminal offence. With 30 listed buildings in a district of tens of thousands of homes, the odds are that yours isn't one, but it's worth checking.
There is one conservation area in Catford, and Historic England hasn't published its name. Inside a conservation area the council has more say over what's visible from the street, such as front extensions, windows, roofs and some demolition, and some of the usual automatic rights to build are reduced. Ask Lewisham Council whether the house falls inside the boundary, because with the name unpublished you can't easily look it up.
The neighbours are busy. Lewisham Council received 1,082 planning applications from Catford between April 2024 and June 2026, 504 of them in the last twelve months. That doesn't tell you how many were approved, and nothing here measures that. It does tell you that a mostly pre-1930 district is constantly asking permission to add rear extensions, loft conversions and other changes.
The ground is the one thing every project here has to deal with. The British Geological Survey puts Catford in class 3 for shrink-swell clay, and every part of the area reaches that class. Clay like that swells when it's wet and shrinks when it dries, and trees nearby make it worse. That doesn't stop anyone building. It means the foundations will probably need to go deeper than a builder elsewhere would assume, a structural engineer should see the plans early, and the building control inspector will ask about any nearby trees. The Victorian builders didn't always dig deep enough for this clay, which is why so many old houses here have a crack somebody calls historic.
One listed building in the area is on Historic England's Heritage at Risk register, meaning it's in poor enough condition to be flagged for repair. A building on that register can attract grant money and will certainly attract attention from conservation officers. For someone buying an ordinary house down the road it changes nothing.
Owners of houses outside the conservation area have the usual national rights to build some single-storey rear extensions and loft conversions without a full application, within size limits. Catford's homes are mostly flats, though, and flats don't have those automatic rights. If you're buying a flat in a converted terrace, any extension or loft conversion means a full planning application, and usually the freeholder's permission as well.
Planning freedom here is middling. Heritage rules affect only a small part of Catford, but the clay affects every foundation and most of the homes are flats, which need permission for almost everything.