Ultrafast control of current on the atomic scale is essential for future innovations in nanoelectronics. Extremely localized transient electric fields on the nanoscale can be achieved by coupling picosecond duration terahertz pulses to metallic nanostructures. Here, we demonstrate terahertz scanning tunnelling microscopy (THz-STM) in ultrahigh vacuum as a new platform for exploring ultrafast non-equilibrium tunnelling dynamics with atomic precision. Extreme terahertz-pulse-driven tunnel currents up to 10(7) times larger than steady-state currents in conventional STM are used to image individual atoms on a silicon surface with 0.3nm spatial resolution. At terahertz frequencies, the metallic-like Si(111)-(7 x 7) surface is unable to screen the electric field from the bulk, resulting in a terahertz tunnel conductance that is fundamentally different than that of the steady state. Ultrafast terahertz-induced band bending and non-equilibrium charging of surface states opens new conduction pathways to the bulk, enabling extreme transient tunnel currents to flow between the tip and sample.