The effects of diffusive boundary scattering in the superfluid properties of thin (kF-1d<3/4) He3 films are studied. We assume that surface scattering arises from random, uncorrelated irregularities in the substrate. For a specified surface roughness there is a critical thickness dc below which superfluidity disappears. For d larger than dc superfluidity is only moderately suppressed and certainly within observational limits. As the thickness decreases toward dc we find that the A-phase-like behavior is followed by an intermediate, experimentally accessible region in which the excitation spectrum is gapless. Thus, dirty anisotropic superfluidity is predicted to be realizable in He3 films. We present calculations of the density of states in the gapless regime, and the zero-temperature phase diagram. We briefly discuss the specific heat and other properties of the gapless state.