The effect of hydrostatic pressure on the structure of a bicontinuous microemulsion in the presence of a solid interface has been studied by X-ray reflectometry and compared to the bulk behavior determined by small-angle X-ray scattering. Surface induced lamellar ordering is observed close to the hydrophilic interface, which persists upon compression. The lamellar domains are compressed, but the correlation length of lamellar order does not change with pressure. SAXS measurements on the bulk microemulsion revealed an increased order upon pressurization. Although pressure can cause the formation of highly ordered lamellar phases from ordered bicontinuous cubic phases, such a scenario is not observed for the disordered analogue studied here. High pressure increases the stiffness of the interfacial surfactant layer, but this is not sufficient to overcome the loss in conformational entropy that would result from a transition to an ordered lamellar phase. Possible technological and biological implications of our results are briefly discussed.