Yes, you can, but in Rye the building often gets a vote. Over three hundred properties are listed. A handful carry the top grades. The neighbours still put in planning applications by the hundred. You can extend here; doing it fast is another matter.
Historic England's register lists 317 buildings in Rye. Five are Grade I, 11 are Grade II* and 301 are Grade II. Listing covers the whole building, inside and out, so work that would be routine elsewhere, like new windows, a moved wall or a different roof covering, can need listed building consent here as well as, or instead of, planning permission. Doing listed-building work without consent is a criminal offence, not a telling-off. With 317 of these in a town of Rye's size, the odds that the house you're viewing, or the one next to it, is on the list are worth checking before your architect starts on anything ambitious.
There are also 8 scheduled monuments in the area, which is the strictest tier of protection. That mainly matters if your land borders one, where even digging a foundation can need consent from a different authority altogether.
The good news is that people do get work done here. The planning records show 509 applications in the Rye area between March 2024 and June 2026, and 230 in the twelve months to June 2026. That's evidence that people extend, alter and argue with the planners here as a matter of routine. It doesn't tell you how many were approved, and nobody should read it as if it did.
What this means in practice is that permitted development rights, which let you do some extensions without a full application, don't apply to listed buildings in the same way. Their curtilage can be restricted too. For an unlisted post-war semi at the edge of town, the usual national rules are the starting point. For anything listed, or right next to a monument, assume consent is needed and ask the council before you ask a builder. A pre-application conversation costs far less than knocking out work you did without permission.
In Rye, your freedom to alter a home is limited: fine on the modern streets, hard work anywhere near the old ones. Budget for time, drawings and patience, and leave the house's history alone.