A 2-cell embedding of a hypergraph on a closed surface is called regular if its automorphism group acts regularly on its flags. This paper gives an enumeration of regular hypermaps with the automorphism group \({\overline{G}}=\hbox {PSL}(3,p)\) for a prime p and hyperface-valency at least 6. In this paper, every hypermap will be represented by \(\mathcal {H}=\mathcal {H}({\overline{G}}; {\overline{t}} , {\overline{r}}, \overline{\ell })\) for some involutions \({\overline{t}} , {\overline{r}}\) and \(\overline{\ell }\) . Up to the hypermap isomorphism, we list \({\overline{t}} \) , \({\overline{r}}\) and the number n(t, r) of hypermaps (that is the number of \(\overline{\ell }\) ) for given \({\overline{t}} \) and \({\overline{r}}\) . In principle, \(\overline{\ell }\) can be explicitly written but we shall not do that, as it is very cumbersome.
