  We report on sub-cycle terahertz light-field emission of electrons from
tungsten nanotips under extreme conditions corresponding to a Keldysh parameter
$\gamma_K\approx10^{-4}$. Local peak THz fields up to 40~GV/m are achieved at
the apex of an illuminated nanotip, causing sub-cycle cold-field electron
emission and acceleration in the quasi-static field. By simultaneous
measurement of the electron bunch charge and energy distribution, we perform a
quantitative test of quasi-static Fowler-Nordheim tunnelling theory under field
conditions that completely suppress the tunnel barrier. Very high bunch charges
of $\sim10^6$ electrons/pulse are observed, reaching maximum energies of
3.5~keV after acceleration in the local field. The energy distribution and
emission current show good agreement with Fowler-Nordheim theory even in this
extreme field regime. Extending this model to the single-shot regime under
these conditions predicts peak electron distributions with a spectral purity of
$10^{-4}$. THz field-induced reshaping and sharpening of the nanotip is
observed, reducing the tip radius from 120~nm to 35~nm over roughly $10^9$ THz
shots. These results indicate THz-driven nanotips in the extreme field limit
are promising electron sources for ultrafast electron diffraction and
microscopy.
