Yes, and heritage rules are unlikely to be what stops you. Bordon has almost no listed buildings. There is one conservation area. Your neighbours file planning applications all the time. The thing to check is the ground under the extension, not the history around it.
Bordon has two listed buildings, both Grade II, and none at Grade I or Grade II*. Listing means almost any change to the building, inside or out, needs listed building consent on top of planning permission, and working without it is a criminal offence rather than a paperwork slip. With only two in the area, the odds that your house is one of them are small, and the seller's solicitor will know if it is. For most buyers, listing is somebody else's problem.
There is one conservation area, Headley Mill. Inside a conservation area, some work that would otherwise go ahead without an application needs planning permission, and demolition, tree work and changes to the front of a house get closer scrutiny. Outside it, which covers most of Bordon, those extra controls don't apply.
Two scheduled monuments sit in the area. Scheduling is the strictest protection there is, and it matters mainly if your land runs up to one, because any work affecting a monument needs separate consent from the government.
The neighbours are extending. Between March 2024 and June 2026, 594 planning applications were lodged for the area, 262 of them in the last twelve months. That is the planning system in regular use by people making ordinary changes to ordinary houses. It says nothing about which ones were approved, so check the council's planning register for homes like yours on your street before you assume your design will go through.
The ground needs more thought than the heritage. The British Geological Survey places Bordon in shrink-swell class 3, and every postcode in the area reaches that class. Shrink-swell means clay soils that swell when wet and shrink when dry, moving whatever is sitting on them. For an extension, that usually means deeper foundations and a structural engineer's input before the design is fixed, and it's worth asking any trees near the footprint how thirsty they are. None of this is a reason not to build. It's a reason to budget for the foundations before you budget for the bifold doors.
In practice, most homes outside the Headley Mill conservation area and not listed will have normal permitted development rights, so modest extensions and loft conversions can often go ahead without a full application, subject to the national size limits. Anything bigger, or anything inside the conservation area, goes through East Hampshire's planning process. A lawful development certificate costs little and settles the question before the builders arrive.
Your freedom to extend in Bordon is middling: the heritage controls barely register, but the clay underneath means every extension starts with the foundations.