Measurements of the far-infrared absorption spectra of PbSe nanocrystals and nanorods are presented. As the aspect ratio of the nanorods increases, the Fröhlich sphere resonance splits into two peaks. We analyze this splitting with a classical electrostatic model, which is based on the dielectric function of bulk PbSe but without any free-carrier contribution. Good agreement between the measured and calculated spectra indicates that resonances in the local field factors underlie the measured spectra.