Yes, extending a protected structure is often possible, but it will almost always require planning permission and careful design that respects the building's special interest. Extensions are usually assessed on how well they are sited, scaled and detailed relative to the original structure, and conservation officer input is normally expected.
Extending a protected structure is not prohibited outright, but it always requires planning permission, since the normal exempted development rights for house extensions under Schedule 2 of the Planning and Development Regulations 2001 do not apply to protected structures. Any proposal, however small, needs a full planning application, usually accompanied by a heritage or conservation impact statement explaining how the design responds to and protects the structure's special interest.
Planning authorities generally assess extensions to protected structures on the basis of reversibility, subordination and sensitivity: a well-designed extension should be clearly identifiable as a later addition rather than disguised as original fabric, should be subordinate in scale and positioned so as not to dominate or obscure the principal elevations, and should ideally avoid the loss of historic fabric where it connects to the existing building. Contemporary design is not automatically rejected, and many conservation officers favour honest, high-quality modern extensions over pastiche additions that copy historic detailing inaccurately.
Because outcomes depend heavily on the specific structure, its setting, and the local authority's Development Plan policies, it is strongly advisable to engage a conservation architect and consult the planning authority's conservation officer at pre-application stage before committing to a design, since this can save significant time and cost compared to submitting an application that has not accounted for heritage concerns.