We have planning permission for a 2-storey 3m-deep rear extension to our detached house. Can we extend it with a further 4m single-storey extension under Permitted Development?

Extensions under Permitted Development (PD) rules can only be built from the original back wall of the building. You might be able to make an application under PD as long as the additional extension does not attach to the approved extension. First, I would check that PD rights have not been removed as part of the approval process for your original application. If you still have PD rights, I would then ask the Planning Officer for advice on this issue.