You can, but in Rotherhithe more of your plans than usual will need someone else's permission first. Hundreds of listed buildings stand in and around the district. Two dozen conservation areas cover the surrounding streets. The ground is clay that moves. Extend here and plan for paperwork, drawings and a good deal of patience.
Historic England lists 318 buildings within the area drawn around Rotherhithe, a box that also reaches across the river. Twelve are Grade I, 13 are Grade II* and 293 are Grade II. Listing covers the whole building, inside and out, so on a listed home even a new window, a moved wall or a replacement door needs listed building consent on top of any planning permission. Doing the work without consent is a criminal offence, and an owner can be made to undo it.
The same area holds 25 conservation areas, among them Edward III's Rotherhithe and St Mary's Rotherhithe. Inside a conservation area the council takes a closer interest in anything visible from the street, from windows and roofs to demolition, and some changes that would go through automatically elsewhere need an application here. Living in Edward III's Rotherhithe means the person checking your rear extension is, in a sense, working on behalf of a long-dead king.
Eight scheduled monuments sit inside the box as well, including the pumping engine house for Brunel's Thames tunnel. Scheduling is the strictest form of protection, and it mostly matters to anyone whose land or building work comes close to one.
People do get things approved. Southwark logged 816 planning applications across Rotherhithe's postcodes between April 2024 and June 2026, and 343 of them in the most recent twelve months. Those figures count how many people tried, and say nothing about how many succeeded, but they do show a neighbourhood that keeps knocking on the planning office door. Your idea for the back of the house has probably been through Southwark's planning team before.
The ground needs the same attention as the paperwork. The British Geological Survey puts the whole area in class 3 for shrink-swell clay, a published hazard class meaning the clay swells when wet and shrinks when dry enough to move shallow foundations. For an extension that means deeper foundations, a structural engineer's design and a building control officer who will want to see both. It is a standard extra cost, and nothing to lose sleep over.
Seven listed buildings in the area are on Historic England's Heritage at Risk register, which means they are in poor condition. Repairs to them may qualify for grant support and will certainly get close scrutiny. Unless you are buying one of the seven, it has no bearing on the house next door.
Most homes here are flats, and flats have no householder permitted development rights, so almost any external change to one needs planning permission and the freeholder's agreement as well. Houses in conservation areas have reduced rights, and listed houses in effect have none. Look the address up on Southwark's conservation area map and the national heritage list while you are still choosing, because whether a house is listed decides the job more than how much space you have.
Rotherhithe gives owners very little freedom to change their homes, and anyone who wants to add a storey or a glass box at the back should expect a long negotiation before they get the builders in.