The packing chromatic number $\pcn(G)$ of a graph $G$ is the smallest integer $k$ such that its set of vertices $V(G)$ can be partitioned into $k$ disjoint subsets $V_1$, \ldots, $V_k$, in such a way that every two distinct vertices in $V_i$ are at distance greater than $i$ in $G$ for every $i$, $1\le i\le k$. For a given integer $p \ge 1$, the generalized corona $G\odot pK_1$ of a graph $G$ is the graph obtained from $G$ by adding $p$ degree-one neighbors to every vertex of $G$. In this paper, we determine the packing chromatic number of generalized coronae of paths and cycles. Moreover, by considering digraphs and the (weak) directed distance between vertices, we get a natural extension of the notion of packing coloring to digraphs. We then determine the packing chromatic number of orientations of generalized coronae of paths and cycles.