One of the best websites for all advanced water properties is a website maintained by Dr. Martin Chaplin http://www1.lsbu.ac.uk/water/physical_anomalies.html#Pvisc
Keep in mind that viscosity is a really complicated subject and water is among the unique solvents. In general viscosity of liquids increases with pressure. At pressure extremes, let us say, 100,000 psi, many liquids become solids. Water also becomes a solid at really really high pressures. So when you talk about viscosity, you should talk about what pressure and what temperature.
From the website:
Water's pressure-viscosity behavior [534, 2890] can be explained by
the increased pressure (up to about 100-200 MPa) causing deformation,
so reducing the strength of the hydrogen-bonded network, which is also
partially responsible for the viscosity. This reduction in
cohesiveness more than compensates for the reduced void volume. It is
thus a direct consequence of the balance between hydrogen-bonding
effects and the van der Waals dispersion forces  in water; with
hydrogen-bonding prevailing at lower temperatures and pressures. At
higher pressures (and densities), the balance between hydrogen-bonding
effects and the van der Waals dispersion forces is tipped in favor of
the dispersion forces and the remaining hydrogen bonds are stronger
due to the close proximity of the contributing oxygen atoms .
Viscosity, then, increases with pressure.
Second question on the viscosity of gases
The textbooks seems to be talking about a limited range of P. Viscosity of gases is dependent on very low or very high pressures. However, for ideal gases and normal pressures, it is only dependent on the temperature. Yes, it is quite counterintuitive. Also viscosity of gases increases with temperature. This is another surprise. https://en.wikipedia.org/wiki/Viscosity#Pure_gases
You can try posting this second query in Physics forum. I don't know the derivation of that Wikipedia expression.