  We report measurements of the shear viscosity $\eta$ in water up to
$150\,\mathrm{MPa}$ and down to $229.5\,\mathrm{K}$. This corresponds to more
than $30\,\mathrm{K}$ supercooling below the melting line. The temperature
dependence is non-Arrhenius at all pressures, but its functional form at
$0.1\,\mathrm{MPa}$ is qualitatively different from that at all pressures above
$20\,\mathrm{MPa}$. The pressure dependence is non-monotonic, with a
pressure-induced decrease of viscosity by more than 50 % at low temperature.
Combining our data with literature data on the self-diffusion coefficient
$D_\mathrm{s}$ of water, we check the Stokes-Einstein relation which, based on
hydrodynamics, predicts constancy of $D_\mathrm{s} \eta/T$, where $T$ is the
temperature. The observed temperature and pressure dependence of $D_\mathrm{s}
\eta/T$ is analogous to that obtained in simulations of a realistic water
model. This analogy suggests that our data are compatible with the existence of
a liquid-liquid critical point at positive pressure in water.
