For most houses, yes, and fairly freely. The heritage footprint here is small: 26 listed buildings, a single conservation area, and no scheduled monuments inside the town. Unless your home is one of the listed few or sits inside that one protected pocket, the ordinary rules apply and the ordinary extensions are on the table. The constraint in Hythe is the exception rather than the rule, which is not something every waterside town can say.
All 26 of the town's listed buildings carry the lowest of the three grades, grade two, which covers the great bulk of listed homes across the country and is a long way from untouchable. Listing still bites, though. It applies to the whole building, inside and out, so swapping the windows, moving an internal wall, or bolting on an extension needs listed-building consent on top of anything else, and "it was only a small change" is not a defence the system recognises. If you buy one of these, budget for slower decisions and specialist advice. What you cannot do is treat it as an ordinary house that happens to have character.
The town also holds one conservation area, covering old Hythe. Inside that boundary the usual freedoms shrink: work that would be automatic elsewhere can need permission, demolition is controlled, and even trees can carry protection, so the quick render-and-cladding facelift tends to run into a consent you did not see coming. Step outside the boundary and none of it applies, which describes most of the town.
In practice Hythe splits cleanly down that line. In the post-war streets that make up the bulk of the place, permitted development does most of the heavy lifting: a rear extension, a loft conversion or a porch usually proceeds without a full application, within the standard limits. Move into the conservation area or buy a listed house and you cross into consent territory, where the answer is still often yes but the route is longer and the paperwork is real. The town plainly keeps building either way, with roughly 150 planning applications going through in the last year alone.
The verdict lands mid-scale, which for a would-be extender is good news. Outside a couple of small protected patches, Hythe leaves you largely free to alter your home. Buy in the old core and you trade that freedom for the setting. Buy anywhere else and the only thing between you and a bigger kitchen is a builder and a budget.