Yes, and Colden Common leaves you more freedom to do it than most villages of its vintage. Only eleven buildings in the entire place carry a listing. There are no conservation areas draped over the streets to police the details. That leaves the ordinary homeowner with the ordinary permitted-development freedoms and very little extra heritage machinery to fight through. The obstacles here are the ones every house faces, not a special local layer on top.
The listed stock is small and specific. Eleven buildings are listed, and they sit at the grade II level, the everyday tier of protection rather than the exceptional one. Listing matters enormously if you happen to own one of those eleven, because it reaches inside the house as well as outside it. Windows, internal walls, chimneys, even the fabric of the thing can need consent before you touch them, and the bar for changes is set to preserve character rather than to suit a kitchen-diner. But that is a rule for a handful of addresses. For the overwhelming majority of homes in the village, listing is somebody else's problem, and the protected buildings act more as scenery than as constraint.
In practice, most owners here work inside the same permitted-development envelope as anyone else in England, the rear extension, the loft conversion, the outbuilding, the sort of thing that can go ahead without a full application if it stays within the size and height limits. Step outside that envelope, or own one of the listed properties, and you are into formal consent territory, which is a process rather than a refusal. There is one boundary worth checking before you draw up plans. Part of the parish falls inside the South Downs National Park, where permitted-development rights tend to be tighter, so the first question for any plot is whether it sits inside that line or outside it. That single fact can change what you are allowed to do without asking, and it is worth confirming with the local planning authority before an architect starts charging by the hour.
So can you extend a house in Colden Common? For most homes, yes, and with a good deal of latitude. The heritage footprint is light, the conservation-area count is zero, and unless you have bought one of the eleven listed buildings or a plot inside the national park boundary, your freedom to alter and extend is wide. Check the boundary, then build.