  The collective motion of membrane lipids over hundred of nanometers and
nanoseconds is essential for the formation of submicron complexes of lipids and
proteins in the cell membrane. These dynamics are difficult to access
experimentally and are currently poorly understood. One of the conclusions of
the celebrated Saffman-Debr\"uck (SD) theory is that lipid disturbances smaller
than the Saffman length (microns) are not affected by the hydrodynamics of the
embedding solvent. Using molecular dynamics and coarse-grained models with
implicit hydrodynamics we show that this is not true. Hydrodynamic interactions
between the membrane and the solvent strongly enhance the short-time collective
diffusion of lipids at all scales. The momentum transferred between the
membrane and the solvent in normal direction (not considered by the SD theory)
propagates tangentially over the membrane inducing long-ranged repulsive forces
amongst lipids. As a consequence the lipid collective diffusion coefficient
increases proportionally to the disturbance wavelength. We find quantitative
agreement with the predicted anomalous diffusion in quasi-two-dimensional
dynamics, observed in colloids confined to a plane but embedded in 3D solvent.
