In the classic transistor, the number of electric charge carriers--and thus the electrical conductivity--is precisely controlled by external voltage, providing electrical switching capability. This simple but powerful feature is essential for information processing technology, and also provides a platform for fundamental physics research. As the number of charges essentially determines the electronic phase of a condensed-matter system, transistor operation enables reversible and isothermal changes in the system's state, as successfully demonstrated in electric-field-induced ferromagnetism and superconductivity. However, this effect of the electric field is limited to a channel thickness of nanometres or less, owing to the presence of Thomas-Fermi screening. Here we show that this conventional picture does not apply to a class of materials characterized by inherent collective interactions between electrons and the crystal lattice. We prepared metal-insulator-semiconductor field-effect transistors based on vanadium dioxide--a strongly correlated material with a thermally driven, first-order metal-insulator transition well above room temperature--and found that electrostatic charging at a surface drives all the previously localized charge carriers in the bulk material into motion, leading to the emergence of a three-dimensional metallic ground state. This non-local switching of the electronic state is achieved by applying a voltage of only about one volt. In a voltage-sweep measurement, the first-order nature of the metal-insulator transition provides a non-volatile memory effect, which is operable at room temperature. Our results demonstrate a conceptually new field-effect device, extending the concept of electric-field control to macroscopic phase control.