Shen and Zhang (2021) showed that almost periodicity naturally arises in the spectral representation of discrete-time $p$-adic self-similar processes with stationary increments. In this paper, we study several notions of almost periodicity as sample path properties of Banach space-valued $p$-adic sssi processes. We prove that Bohr almost periodicity is equivalent, as a path event, to continuity with respect to the $p$-adic topology. We also show that the corresponding equivalence fails for Weyl and Besicovitch almost periodicity. Finally, we extend the Bohr almost-periodic result to finite-dimensional random fields.