You can, but you'll probably need permission first, and more than once. There are hundreds of listed buildings in the surrounding area. Four conservation areas carry the village's own name. Some of the clay underneath swells and shrinks. The planners are likely to have more say in the design than your architect.
Historic England lists 442 buildings across the wider area. Fourteen are Grade I, 24 are Grade II* and 404 are Grade II. Listing covers the whole building, inside and out, so a new window, a knocked-through wall or a changed staircase in a listed home normally needs listed building consent on top of any planning permission. Grade II is the most common grade, and it's still listed. Buyers who assume it's the minor version tend to find out the hard way.
The surrounding area also has 17 conservation areas, and four of them are East Peckham's own: Bullen Corner, Little Mill, Roydon and Snoll Hatch. Inside a conservation area the council looks harder at anything that changes how a street looks from the pavement, and some work that elsewhere needs no application needs one here. Check the council's conservation area maps before you check the kitchen.
There are six scheduled monuments in the area too. That's the strictest protection there is, and it matters mainly if your land runs up to one.
The neighbours are clearly extending anyway. There were 2,909 planning applications across the area between April 2024 and June 2026, including 1,251 in the last twelve months. Plenty of people here are asking permission to change their homes. What those numbers can't tell you is how many were approved, so read the decisions on your own street before assuming yours will go through.
The ground matters as much as the paperwork. The national geological survey's shrink-swell hazard reaches class 3 in parts of the area, and about 22% of it is at that level or above. Clay at that level swells when it's wet and shrinks when it's dry, so extensions may need deeper foundations and a structural engineer's view before anyone digs. That's a cost to budget for early. The real risk is finding out after the builder has started.
Before buying, find out whether the house is listed, whether it sits in a conservation area and what the ground is like. If the answer to all three is no, ordinary permitted development may cover a modest extension. If any of them is yes, assume every change needs an application, and ask the council before you commission any drawings.
East Peckham is a poor choice if you plan to change a house much. Buy one that's already the size you need.