A common feature of water close to nonpolar particles, at hydrophobic interfaces, and at the surface to its vapor is the signature of ice-like orientated water molecules. By forming ice-like orientations, the number of unsatisfied hydrogen bonds is minimized and thus the energetic penalty is reduced. To what extent these structural motifs correspond to the existence of a real ice layer has been a long-lasting controversial topic, and the view emerges that the solvation layer is only poorly described as being ice. In this work we present new insight into the ice-like conformations of water at a hydrophobic interface by a detailed analysis of molecular dynamics simulations of water at a hydrophobic self-assembled monolayer under pressures from 1 bar up to 10 kbar. We show by analyzing the ice-like order on different length scales that - despite different signatures of ice-like order - the actual density of ice-like structures is negligible at ambient pressure, but significantly grows with increasing pressure contrary to bulk hexagonal ice, which is destabilized by elevated pressure. At 10 kbar, the interfacial water is of significantly ice-like order.