Yes, you can extend a house in New Alresford, and most people do so without much drama. The catch is which house. The town carries around 133 listed buildings and a conservation area laid over its historic core, so a slice of the housing stock comes with a planning officer effectively built in. The great majority of homes, though, are ordinary post-war houses well clear of those controls. Whether extending here is straightforward or a slog comes down almost entirely to the address.
Listing is the strictest layer of all. Around 133 buildings in and around the town are listed, four of them at the higher grade two star and the rest at grade two, with none at the top grade one. If your home is one of them, listed building consent is needed for almost any material change, inside as well as out, from moving an internal wall to swapping the windows. Owning a listed house does not stop you altering it, but it does mean every alteration has to satisfy someone whose job is protecting what makes the building historic, and that tends to close off the cheap and the quick before you start.
Sitting over the top of that is a single conservation area, and it covers New Alresford itself, the old Georgian centre the town trades on. Inside a conservation area the everyday freedoms shrink. Demolition is controlled, trees pick up extra protection, and changes you could make without asking anywhere else often need a formal application here, particularly anything that faces the street. The point of the rules is to keep the Georgian frontages looking like Georgian frontages, which is a large part of why houses here hold their price, so the controls are enforced rather than ornamental.
For everyone else, which is most of the town, the picture is far more relaxed. The bulk of the housing is post-war and sits outside both the listed stock and the conservation core, so the ordinary permitted development rules apply and a rear extension or a loft conversion is routine rather than a fight. The town is no stranger to the paperwork either, working through more than 350 planning applications in the past year alone, so the system is well used to saying yes as often as it says no. The one sensible first move for any particular house is to find out whether it is listed and whether it falls inside the conservation area boundary, because those two answers settle nearly everything that follows.
So the honest verdict lands in the middle. New Alresford gives you decent freedom to extend if you buy a plain modern house, and a genuine run of hurdles if you buy one of the handsome old ones. Character here comes with a permission slip attached, and the town is reasonably even-handed about handing that slip out.