A (face-)primer hypermap is a regular oriented hypermap whose orientation-preserving automorphism group $G$ acts faithfully on the hyperfaces. In this paper, we investigate the primer hypermaps for which $G$ is nilpotent and the hypervertex-valency is a prime $p$. (We call these PNp-hypermaps.) We prove that for any PNp-hypermap, the group $G$ must be a finite $p$-group, and the number of hyperfaces is bounded above by a function of the nilpotency class of $G$. Moreover, we show that for any positive integer $c$, there is a unique PNp hypermap $H_c$ of class $c$ attaining the given bound, and every other PNp hypermap of class at most $c$ is a quotient of H_c.