Wiltshire Council's own published conservation areas layer holds 244 conservation area records for the county, one of which is named Chippenham, reference CA/N/51. The council's open data description states that "the local planning authority have a statutory duty to designate and manage conservation areas under the Planning (Listed Buildings and Conservation Areas) Act 1990", and defines a conservation area as "an area of special architectural or historic interest where the original appearance is preserved or enhanced".
The Planning Portal sets the national rule behind every Chippenham job. It says you will not need to apply for planning permission to erect a new, or alter, maintain, improve or take down an existing fence, wall or gate if all of its conditions are met, among them that it is "next to a highway used by vehicles (or the footpath of such a highway) and it would not exceed one metre in height (from ground level)", or that "it would not exceed two metres in height (from ground level) if elsewhere". Where a Chippenham boundary already exceeds those limits, the condition is that "its height would not be increased".
Those same conditions require, at a Chippenham address, that "no part of the site is a listed building or within the curtilage of a listed building" and that "the right to put up or alter fences, walls and gates has not been removed by an article four direction or a planning condition". The Planning Portal adds that in a Chippenham conservation area "you might need permission take down a fence, wall or gate".
In Chippenham we would explain how you check the conservation boundary, any Article 4 direction and the proposed work before confirming whether permission is needed.