On face value a basement extension could be argued to meet the description of “the enlargement, improvement or other alteration of a dwellinghouse” in Part 1 Class A of the Town and Country Planning (General Permitted Development) Order (GPDO) 2015 and thus that it benefits from permitted development rights provided the proposal meets the criteria set out in Class A.
However, a recent High Court judgment (Eatherley v London Borough of Camden, December 2016) ruled that the basement extension in question could not be lawfully carried out using permitted development rights as the proposal comprised engineering works which were deemed a separate activity in their own right not benefitting from permitted development rights.
The ruling does not mean that all engineering works required to facilitate basement extensions will require planning permission but it does bring the issue into question so you need to be careful when planning an extension of this kind. As a result of the case the London Borough of Camden issued an Article 4 Direction, which comes into force on the 1st June 2017, to remove all permitted development rights for basement extensions. It could be that other Local Authorities follow suit.
The government is currently reviewing the planning law and regulations which relate to basement developments. Until the situation is clarified you are advised to contact a chartered town planning consultant or your Local Planning Authority if planning a basement extension.