Yes, for most houses in Newhaven, with conditions for the ones that fall inside the heritage map. The town has 41 listed buildings and 5 conservation areas. One Grade I and three Grade II* sit in the protected tier. The remaining 37 are Grade II. The rest of the town runs on standard planning rules, which is most of it.
The 41 listed buildings break down into one Grade I, three Grade II, and 37 Grade II. Listing means the building has been formally identified as having historical or architectural significance, and any change that affects its character requires listed building consent on top of normal planning permission. That covers the obvious things like extensions and roof changes, and also the less obvious ones: replacing windows with modern frames, removing internal walls, repainting in non-historic colours, even some kinds of repair work. Grade I and II listings, which here cover four buildings between them, carry tighter constraints than Grade II. Owners of listed properties usually find the consent process slower, more expensive, and more cautious than they expected. The flip side is that the buildings tend to hold their character and their value because of it. The houses that are listed are listed for a reason, and the reason is usually visible from the pavement.
Newhaven sits inside five conservation areas: Piddinghoe, Tarring Neville, Newhaven (Denton), South Heighton, and Newhaven (Church Hill). A conservation area means the local planning authority has identified an area whose character is worth preserving, which is a broader and softer kind of constraint than listing. Permitted-development rights, the rules that let homeowners do small works without applying for planning permission, are usually reduced inside a conservation area. Article 4 directions can remove permitted-development rights altogether for specific things like changing front windows or laying paving. The exact terms vary by area and want checking with the council before any work, including the work a homeowner thinks is routine. There are also two scheduled monuments in the wider area, which are a separate and stricter category, but those affect very few houses directly.
For the practical question of what most buyers actually face, the answer is that the heritage constraints affect a small share of the housing stock. With 41 listed buildings across a town of around 5,900 households, the listed share is a small fraction of the total. The five conservation areas cover defined boundaries that mostly take in the older village centres and historic clusters. A post-war semi on a residential street away from those cores is almost certainly outside any heritage constraint, and an extension or a window change is a normal planning conversation. The Georgian or Victorian terrace inside the conservation boundary is inside one, and the same work is a slower conversation with extra paperwork. Anyone with a specific address can find out which side of the line their property sits on through the council's planning portal before they spend money on architects.
You can extend most houses in Newhaven on the same terms as anywhere else in the country. For the small share that sit inside a conservation area or wear a Grade II badge, the process is slower, the rules are tighter, and the answer is sometimes no.