Yes, in Marchwood you can almost certainly extend, and with far less heritage red tape than most places would throw at you. The great bulk of the village is modern housing that reaches back no farther than the late twentieth century, which means it carries none of the special protections that turn a simple project into a negotiation. There are a handful of listed buildings, all of the gentlest grade, and a single conservation area that covers an old military site rather than anyone's semi. For the ordinary homeowner here, the heritage rulebook is thin. The main obstacle to your extension is far more likely to be your budget than a conservation officer.
On listed buildings, the constraint is real but narrow. Marchwood has around seventeen listed buildings in total, and every one of them sits at grade two, the lowest and most common tier of protection. If you happen to own one, listing bites hard: you need listed-building consent for alterations that affect its character, inside as well as out, and "it was only the windows" is precisely the sort of thing that lands people in front of enforcement. If you own one of the many hundreds of ordinary houses instead, none of that applies, and the odds are overwhelmingly that you do.
Conservation areas are the other thing that can tie hands, and here there is exactly one. It covers the old Royal Naval Armaments Depot rather than a residential street, so for the overwhelming majority of homeowners it is a line on a map that never once touches them. A conservation area can bring extra consents and tighter control over changes you would normally take for granted, but only if your home sits inside its boundary. In Marchwood, almost none do.
In practice, that leaves most owners with the usual national permitted-development freedoms intact: the modest rear extension, the loft conversion, the garden outbuilding, the sort of work that quietly improves a family house without turning into a planning saga. Bigger or bolder schemes will still want full planning permission, the same as anywhere, and the sensible move is always to confirm your own property's status before you draw a single line, because listing and conservation boundaries do not care how reasonable your idea sounds. The starting position here, though, is freedom rather than restriction.
So can you extend a house in Marchwood. For most people, yes, and comfortably: this is a place that scores well precisely because the heritage rules so rarely get in the way. Buy a listed one and the picture tightens considerably. Buy any of the many that are not, and the only real limit is what you are willing to spend.