We study quantum size effects in superfluid He3 films. We consider the thickness d regime kF-1d. In this regime, quantum size effects are most prominent, while several theoretical simplifications are possible. The Gorkov equations for the p-type superfluid are solved in the parallel-plate geometry assuming specularly reflecting walls. A simple decomposition for the pairing potential is used which allows explicit evaluation of the various thermodynamical quantities and the NMR response in the Anderson-Brinkman-Morel state. The transition temperature is monotonically depressed as the thickness decreases, and, contrary to what is found for s-wave pairing, exhibits only weak oscillatory behavior. On the other hand, the shift in the transverse resonance frequency shows characteristic discontinuous jumps. Particular emphasis is placed on experimentally detectable deviations from the bulk behavior. The effects of surface roughness on our results are briefly discussed.