Usually, yes, though in the historic core it takes longer. Sixty-one buildings in the area are listed. One conservation area covers the old village. The clay under every postcode moves with the weather. Outside the core, a post-war semi can be extended much like anywhere else, as long as the foundations are dug deep enough.
Historic England lists 61 buildings in the Bassingbourn area: one at Grade I, none at Grade II*, and 60 at Grade II. Listing covers the whole building, inside and out, so changes that affect its character need listed building consent on top of any planning permission, and that includes some internal work. Grade II is the commonest tier, though it is still listing. A house that has been on the register since before you were born gets a say in your kitchen layout.
The area has one conservation area, Bassingbourn, which covers the historic heart of the village. Within it, demolition, some extensions and changes to how a house faces the street are controlled more tightly, and the council can remove some of the rights to build without planning permission. South Cambridgeshire District Council publishes the boundary, so check the map for the exact house, because buyers routinely get the line wrong by a few doors in both directions.
There is one scheduled monument in the area, the strictest tier of protection in England. Works on or close to it need consent from the national heritage body, and that only affects the handful of properties near it.
Neighbours here extend homes all the time. Planning records list 1,709 applications covering the area between April 2024 and June 2026, 773 of them in the most recent twelve months. That says nothing about how many were approved, but it shows a planning system that is busy here.
The ground is the issue that applies everywhere. The British Geological Survey puts every postcode in the area, 100% of them, in shrink-swell class 3, the highest class found here. Clay soils swell when wet and shrink when dry, and trees make that worse by drawing moisture out. For an extension, that means foundations deeper than standard, care around nearby trees and a structural engineer's input at design stage. The survey says the clay is there and will need planning for, and does not say houses are failing.
In practice, a post-war semi or bungalow outside the conservation area and off the listed register will usually be able to use permitted development for a modest rear extension or loft conversion, subject to the national size limits and a certificate from the council if you want certainty. Anything listed, or inside the conservation area, means a full application and a longer wait. The same single-storey extension can be a quick summer job on a cul-de-sac and a year-long exchange of letters about window frames near the church.
Your freedom to extend in Bassingbourn is middling. The historic core and the clay add rules and extra cost, but the rest of the village is open to the kind of extension people build everywhere else.