Semiconducting carbon nanotubes (CNTs) are attractive as channel material for field-effect transistors due to their high carrier mobility. In this paper we show that a local CNT gate can provide a significant improvement in the subthreshold slope of a CNT transistor compared to back gate switching and provide gate delays as low as 5 ps. The CNT gated CNT transistor devices are fabricated using a two-step chemical vapour deposition technique. The measured transfer characteristics are in very good agreement with theoretical modelling results that provide confirmation of the operating principle of the transistors. Gate delays below 2 ps should be readily achievable by reducing the thickness of the gate dielectric.