Yes, you can extend a house in Higham, and heritage is unlikely to be what stops you. The village has only five listed buildings. No conservation area covers the village itself. The neighbours submit planning applications all the time. Planning isn't the hard part here. The ground is.
Higham has 5 listed buildings: 1 at Grade I, none at Grade II* and 4 at Grade II. Listing protects the whole building, inside and out, so any change to one of those five needs listed building consent on top of any planning permission, and the conversation about uPVC windows will be short. For everyone else, which is nearly everyone, listing doesn't apply. The post-war semis and bungalows that make up most of the village have nothing to protect but their owners' taste.
The planning register shows people extend here all the time. Between April 2024 and June 2026, 1,798 applications were logged across the planning data covering the area, 797 of them in the last twelve months. That's a lot of kitchen extensions, porches and loft dormers for a village this size. If the house two doors down has a new rear extension, that's not proof you'll get one, but it does show the council is used to seeing them.
The ground needs more thought than the paperwork. The British Geological Survey places the whole area, 100% of it, in class 3 for shrink-swell, a published hazard classification. The soil expands when wet and shrinks when dry, and a new extension on shallow foundations can move differently from the house it's attached to. That's a question for a structural engineer and building control, not a reason to walk away: expect deeper foundations than a builder quoting from memory might suggest, and pay particular attention to any trees nearby. The crack that shows up two summers later always costs more than the engineer would have.
In practice, many single-storey rear extensions, loft conversions and outbuildings on an ordinary house may fall within permitted development, which means no full planning application, subject to national size and position limits. Anything larger, anything on the front, or anything on one of the five listed buildings goes through Gravesham's planning process. Check the house's own planning history before you buy, because a previous owner's extension may already have used up part of the allowance, or removed permitted development rights altogether.
Your freedom to extend a house in Higham is above average: the heritage restrictions barely apply, and the main constraint is the ground under the house, not the rules about what sits on top of it.