You can, but in South Lambeth most changes to a home need someone else's permission first. There are hundreds of listed buildings and dozens of conservation areas. The ground is clay that shrinks and swells. Most homes are flats, which have the least freedom of any home to extend. The few terraced houses are where all the ambition, and all the paperwork, ends up.
Historic England lists 383 buildings in and around South Lambeth: 19 at Grade II*, 364 at Grade II and none at Grade I. Listing covers the whole building, inside and out, so knocking through a wall, replacing windows or adding to the back of a listed house needs listed building consent on top of any planning permission. None of the buildings is Grade I, so you won't be living next to a palace, but there are plenty at Grade II.
The wider patch around South Lambeth contains 31 conservation areas, among them South Lambeth Road, Clapham Road and Vauxhall. In a conservation area the council looks harder at anything visible from the street, such as roofs, dormers, front walls and windows, and some work that needs no permission elsewhere needs it here. The council's conservation area maps show which side of a boundary a house sits on, and since a boundary can run down the middle of a street, it is worth checking before you plan anything.
Lambeth's planning register shows 1,059 applications in South Lambeth between April 2024 and June 2026, and 469 in the last twelve months of that period. People here do alter their homes, and in quantity. Whatever you are planning, a neighbour has very likely already been through the forms for something similar.
The British Geological Survey rates the ground as class 3 for shrink-swell, and all of South Lambeth reaches that class. Clay at that level expands when wet and shrinks when dry, so an extension usually needs deeper foundations than it would on firmer ground, especially near trees. A structural engineer's view at design stage is normal here and costs far less than cracks later. For a leasehold flat or house, read the lease for clauses on alterations, because Lambeth has a long history of large estates owning the freehold, back to the Black Prince and the Duchy of Cornwall, and the freeholder may have a say as well as the council.
Two listed buildings in the area are on Historic England's Heritage at Risk register. That means buildings in poor condition whose repair is watched closely and may qualify for grant help. It has no bearing on the ordinary house next door, apart from the scaffolding.
In practice, the freedom to extend depends mainly on what you own. About nine homes in ten here are flats, and flats have far fewer permitted development rights than houses, so even modest changes usually need an application. Terraced houses outside a conservation area keep more of those rights. Inside one, or in a listed house, assume that anything visible needs consent and plan the timetable around the council.
South Lambeth gives you very little freedom to change a home. Buy the home you need now, not one you plan to extend later.