We perform Hochschild homology calculations in the algebro-geometric setting of motivic spectra. Via Betti realization, we recover Bökstedt’s calculation of the topological Hochschild homology of finite prime fields. The motivic Hochschild homology ring contains many torsion classes which arise from the mod-$p$ motivic Steenrod algebra and generating functions on the natural numbers with finite nonempty support.