When two metal nanostructures are placed nanometres apart, their optically driven free electrons couple electrically across the gap. The resulting plasmons have enhanced optical fields of a specific colour tightly confined inside the gap. Many emerging nanophotonic technologies depend on the careful control of this plasmonic coupling, including optical nanoantennas for high-sensitivity chemical and biological sensors, nanoscale control of active devices, and improved photovoltaic devices. But for subnanometre gaps, coherent quantum tunnelling becomes possible and the system enters a regime of extreme non-locality in which previous classical treatments fail. Electron correlations across the gap that are driven by quantum tunnelling require a new description of non-local transport, which is crucial in nanoscale optoelectronics and single-molecule electronics. Here, by simultaneously measuring both the electrical and optical properties of two gold nanostructures with controllable subnanometre separation, we reveal the quantum regime of tunnelling plasmonics in unprecedented detail. All observed phenomena are in good agreement with recent quantum-based models of plasmonic systems, which eliminate the singularities predicted by classical theories. These findings imply that tunnelling establishes a quantum limit for plasmonic field confinement of about 10(-8)λ(3) for visible light (of wavelength λ). Our work thus prompts new theoretical and experimental investigations into quantum-domain plasmonic systems, and will affect the future of nanoplasmonic device engineering and nanoscale photochemistry.