We use recent observations of the H i mass function to constrain galaxy formation. The data conflict with the standard model where most of the gas in a low-mass dark matter halo is assumed to settle into a disc of cold gas that is depleted by star formation and supernova-driven outflows until the disc becomes gravitationally stable. Assuming a star formation threshold density supported by both theory and observations, this model predicts H i masses that are much too large. The reason is simple: supernova feedback requires star formation, which in turn requires a high surface density for the gas. Heating by the ultraviolet background can reduce the amount of cold gas in haloes with masses <109.5 h−1 M⊙, but is insufficient to explain the observed H i mass function. A consistent model can be found if low-mass haloes are embedded in a pre-heated medium, with a specific gas entropy ∼10 keV cm2. In addition, such a model simultaneously matches the faint-end slope of the galaxy luminosity function without the need for any supernova-driven outflows. We propose a pre-heating model where the medium around low-mass haloes is pre-heated by gravitational pancaking. Because gravitational tidal fields suppress the formation of low-mass haloes while promoting that of pancakes, the formation of massive pancakes precedes that of the low-mass haloes within them. We demonstrate that the progenitors of present-day dark matter haloes with M≲ 1012 h−1 M⊙ were embedded in pancakes of masses ∼5 × 1012 h−1 M⊙ at z∼ 2. The formation of such pancakes heats the gas to a temperature of 5 × 105 K and compresses it to an overdensity of ∼10. Such gas has a cooling time that exceeds the age of the Universe at z≲ 2, and has a specific entropy of ∼15 keV cm2, almost exactly the amount required to explain the stellar and H i mass functions.