We show that the optical response of ultrathin metallic films of finite lateral size and thickness can feature peculiar magneto-optical effects resulting from the spatial confinement of the electron motion. In particular, the frequency dependence of the magnetic permeability of the film exhibits a sharp resonance structure shifting to the red as the film aspect ratio increases. The films can also be negatively refractive in the IR frequency range under proper tuning. We show that these magneto-optical properties can be controlled by adjusting the film chemical composition, plasmonic material quality, the aspect ratio, and the surroundings of the film.