A set-valued mapping \(M\) from a Hausdorff topological vector space \(E\) into a normed vector space \(F\) is called directionally pseudo-Lipschitz if the function \(\Delta _{M}:F\times F\rightarrow {\mathbb R}\) given by \(\Delta _{M}(x,y) =d(y,M(x))\) is directionally Lipschitz in the sense of [\textit{R. T. Rockafellar}, Proc. Lond. Math. Soc. (3) 39, 331--355 (1979; Zbl 0413.49015)]. The authors establish a geometric characterization of these mappings and study their tangential regularity by using three suitable notions of tangent cones introduced in the paper.