The Cherenkov fields of a proton, and a neutron, moving with a relativistic velocity in a spatially dispersive medium are studied in the rest frame of the particle. The model of the medium used is typical of the behaviour of a dielectric near an exciton transition, and includes as a special case a screening medium like an isotropic plasma. The Fourier integral for the field of a proton is shown to split up into three integrals, each of which is identical to that in an ordinary medium but for a weight factor dependent on the frequency of the Fourier component. Each of these integrals is associated with one mode of Cherenkov emission, with its own threshold. The motion of the charge gives rise to three coaxial diffuse circular field cones with an azimuthally symmetric intensity distribution. The output of photons in each mode is evaluated. The field and output of a relativistic neutron are also evaluated for different orientations of the magnetic moment of the neutron relative to the direction of motion. It is shown that there are only two cones in this case, consistent with the fact that magnetic sources cannot excite the longitudinal plasma mode in a medium which is spatially dispersive only in its electrical properties.