For most houses in Eastleigh, yes, you can extend, subject to the ordinary rules that apply anywhere in England. The town carries only a modest amount of protected heritage. There are a handful of listed buildings and a small number of conservation areas, and both land on a minority of addresses. The heaviest restrictions apply to specific homes rather than the town at large. So the honest answer is that the constraints are real but they are the odd address here, not the default.
Start with the listed buildings, because those are the ones where extending gets genuinely hard. Eastleigh has 34 listed buildings in total, and all but one of them sit at grade two, the everyday tier of listing, with a single property carried up at grade two-star. If you own one of these, the rules are strict and they apply inside as well as out. Changing windows, knocking through a wall, altering the roof line, even some things that feel cosmetic, can need listed building consent before you touch them. It is a small club, but membership comes with a planning officer attached, and buying into it by accident is an expensive way to learn the rule.
Conservation areas are the softer constraint that catches a wider net. The town has three of them, at Bassett Green Village, Bishopstoke and Ethelburt Avenue, and living inside one changes the calculation without freezing it. Demolition tends to need permission, some of the permitted-development freedoms an ordinary house enjoys can be trimmed back, and the appearance and materials of what you build get more scrutiny than they would two streets outside the boundary. It is control rather than prohibition. You can still improve the house, you just do it with the character of the area looking over your shoulder.
For the great majority of Eastleigh homes, which are neither listed nor inside one of those three areas, the picture is straightforward. National permitted-development rights cover a good deal of what most families actually want, a rear extension, a loft conversion, a bit more kitchen, often without a full planning application, as long as you stay inside the standard limits. The town is plainly used to people building. Something over a thousand planning applications were lodged in the past year, so the process is well-worn rather than exotic, and the officers have seen your extension before. The sensible move is always to confirm your specific status first, because whether a house is listed or in a conservation area is not something you can tell by looking at the brickwork.
So can you extend a house in Eastleigh. For most buyers, comfortably, within the normal national rules. For the minority who land a listed building or a home inside a conservation area, expect to ask permission and expect the answer to come with conditions. Middling freedom overall: not a town that says no by default, but not one where you should assume yes without checking the address.