Yes, and with fewer obstacles than in most villages its age. Galleywood has no conservation area. It has only a handful of listed buildings. Most of the houses are post-war and carry no heritage restrictions at all. The thing most likely to push back on your plans is the clay under them.
Galleywood has 18 listed buildings: none at Grade I, 1 at Grade II* and 17 at Grade II. Altering any of them, inside or out, needs listed building consent on top of any planning permission, and that covers changes like windows, walls and fireplaces. Outside those 18, there is no conservation area anywhere in Galleywood and no scheduled monument. The rest of the village was mostly built from the 1920s onwards, much of it as council housing in the 1960s and 1970s with private estates alongside, and a house from those decades carries no heritage weight at all.
Plenty of people are already doing it. The planning records for the area around Galleywood hold 4,100 applications between April 2024 and June 2026, with 1,714 in the latest twelve months. That counts applications made, and says nothing about how many were approved. Galleywood's main architectural style is "with extension".
The British Geological Survey puts the area in shrink-swell class 3 for clay, and 100% of Galleywood's postcodes reach that class. Clay of this kind swells when it is wet and shrinks when it dries, so foundations for a new extension may need to go deeper than a builder's standard quote assumes, especially near trees. That is a matter for a structural engineer and a building control officer, and it should be budgeted for. In Galleywood, the ground has firmer views on your extension than the planning department does.
Many houses here can be extended under national permitted development rules, without a full planning application, as long as the work stays within the size and position limits those rules set. Flats do not get those rights, and listed buildings always need consent. Some estates had permitted development rights removed by a condition when they were first approved, so ask Chelmsford City Council whether the house you want is one of them, while it is still a house you are only thinking about. That one answer decides whether the rear extension needs a planning application or only building regulations.
Freedom to extend is one of Galleywood's real strengths: there is very little heritage in the way, so the clay and your budget are the main limits.