Yes, you can extend in Whitchurch, as long as your house isn't one of the old ones everyone photographs. Most homes here are post-war. The historic centre is a conservation area. Several dozen buildings are listed. The neighbours send in planning applications all the time. How old the house is decides more than how ambitious the plans are.
Whitchurch has 54 listed buildings. None is grade I, 3 are grade II* and 51 are grade II. Listing protects the whole building, inside and out, not just the front that looks good on the postcard, so any change that affects its character needs listed building consent on top of any planning permission. That covers a lot more than extensions. Owners who assumed the protection stopped at the front door tend to find out otherwise after they've started the work.
There are two conservation areas, Whitchurch and Tufton. Inside them, the council takes a closer interest in anything visible from the street, and some work that would go through without an application elsewhere needs one here. A corner of the North Wessex Downs National Landscape also reaches into about 8% of the area, and protected landscapes tighten the rules too. A house in one of these areas can still grow, but it'll need plans drawn up with the planning officer in mind.
People do extend here, and in numbers. Between April 2024 and June 2026 the area logged 294 planning applications, 126 of them in the last twelve months. For a town of this size, that's a lot of people who looked at their house and decided it needed another room. Whether each one succeeded is another matter, but the volume shows extending here is common, not a fight against the council.
In practice, the answer depends on the address. A post-war semi or a detached house outside the conservation areas and the protected landscape will often get a modest rear extension or loft conversion through permitted development without a full application, subject to the national limits. Inside a conservation area or the protected landscape, expect more of that work to need permission. On a listed building, assume nothing is allowed until the council says it is, and budget for an architect who has done listed work before. Check the listing and the conservation boundary before you make an offer, not after the builder has quoted.
Whitchurch sits in the middle for freedom to build: the post-war streets give you room and the old centre gives you paperwork. Choose the house for the extension you want, not the other way round.