The spin susceptibility of an exchange enhanced metallic film is studied with the objective of determining how the paramagnetic properties are modified by the surfaces and the finite size of the system. In contrast to the usual treatments, an exchange interaction of finite range λ, comparable to the Fermi wavelength λF, is used. The exchange interactions are taken into account within a mean field theory approximation.The spin susceptibility is explicitly evaluated for a model in which the boundaries are treated as infinitely high potential barriers. It is found that the results depend sensitively on the value of λ. In particular, the large quantum oscillations in the surface magnetization found by earlier workers are significantly reduced in magnitude when a realistic value of the range is used. A generalized Stoner criterion determining the ferromagnetic instability in a film of thickness L is also derived from the spin susceptibility. The critical temperature TcF for a film is found to be depressed below the bulk value TcB for two physically distinct reasons. The first is referred to as a density-of-states effect and gives a depression proportional to L−1; in the absence of density-of-states effects the shift is proportional to L−2.