  The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest
integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$,
$i\in \{1,\ldots,k\}$, where each $V_i$ is an $i$-packing. In this paper, we
consider the packing chromatic number of several families of
Sierpi\'{n}ski-type graphs. While it is known that this number is bounded from
above by $8$ in the family of Sierpi\'{n}ski graphs with base $3$, we prove
that it is unbounded in the families of Sierpi\'{n}ski graphs with bases
greater than $3$. On the other hand, we prove that the packing chromatic number
in the family of Sierpi\'{n}ski triangle graphs $ST^n_3$ is bounded from above
by $31$. Furthermore, we establish or provide bounds for the packing chromatic
numbers of generalized Sierpi\'{n}ski graphs $S^n_G$ with respect to all
connected graphs $G$ of order 4.
