  In this paper we investigate the qualitative behaviour of the pressure
function beneath an extreme Stokes wave over infinite depth. The presence of a
stagnation point at the wave-crest of an extreme Stokes wave introduces a
number of mathematical difficulties resulting in the irregularity of the free
surface profile. It will be proven that the pressure decreases in the
horizontal direction between a crest-line and subsequent trough-line, except
along these lines themselves where the pressure is stationary with respect to
the horizontal coordinate. In addition we will prove that the pressure strictly
increases with depth throughout the fluid body.
