Yes, in most of the village, although it gets harder near the green. Fewer than twenty buildings in the area are listed. One conservation area covers the old core. Neighbours file a few hundred planning applications a year. The planning office is manageable. The clay under the house is the harder part.
There are 18 listed buildings in the area, none at Grade I, one at Grade II* and 17 at Grade II. Listing covers the whole building, inside and out, so changing a window, a staircase or an old internal wall on a listed house needs consent on top of any planning permission. Buyers of listed homes tend to find this out after they have picked the kitchen. Eighteen is a small number for a village this old, so most buyers will never own one.
Fernhurst has one conservation area, named after the village, around its historic heart. Inside it, the planning authority pays more attention to anything that changes how a house looks from the street, including materials, rooflines and demolition. The brief does not show any extra restrictions on top of that, so check the specific address rather than assume either way. The parish also lies inside the South Downs National Park, and it is worth asking the planning authority early whether that affects what you want to build.
Local planning records show 753 applications between April 2024 and June 2026, with 321 in the last twelve months. That is a lot of home improvement for a village that complains about change. The number of applications does not tell you how many were approved. It does show that people here extend, convert and alter their homes regularly, and that you will not be the first on your road to apply.
The ground is the more serious constraint. The British Geological Survey's shrink-swell hazard map puts the area at class 3, and every postcode reaches that level. Fernhurst sits on valley clay, which shrinks in dry summers and swells in wet winters, whatever your planning permission says. For an extension, that means deeper foundations and a structural engineer involved early, not a builder working from memory. It is a known issue with a known fix, and the fix costs money.
In practice, a rear extension on one of the post-war streets away from the old core is the straightforward case: check whether it falls under permitted development, and apply if it does not. A listed house, or anything inside the conservation area, needs consent before work starts, and that takes longer. Either way, plan the foundations around the clay before you plan the extra room.
For most homes in Fernhurst, extending is quite achievable. Heritage rules affect relatively few houses, and the clay is an engineering cost rather than a reason for refusal.