In an era of new developments in nanomaterials analysis enabled by the unprecedented spatial and energy resolutions of electron energy-loss spectroscopy in the scanning transmission electron microscope (STEM-EELS), it remains that the vast majority of works concern collective or single-particle excitations that are well described by the electrostatic approximation, which neglects retardation and magnetic field effects. Here we demonstrate a simple case in which that approximation is fundamentally inadequate. When the beam energy is above the Cherenkov threshold and the geometric dimensions of the nanomaterial sample are on the order of the wavelength of light in the material, spatial variations in low-loss (less than or similar to 5 eV) spectral maps from guided light modes may be observed. We demonstrate such observations for amorphous silicon disks and offer an interpretation of the results based on the waveguide modes of a cylinder. We also demonstrate explicitly that spatial variations from waveguide modes are manifest in analytic models for the especially simple geometry of a STEM beam penetrating a dielectric ribbon. We discuss how these modes relate to those that have been observed more generally in dielectric nanomaterials.