  We introduce a continuous-time quantum walk on an ultrametric space
corresponding to the set of p-adic integers and compute its time-averaged
probability distribution. It is shown that localization occurs for any location
of the ultrametric space for the walk. This result presents a striking contrast
to the classical random walk case. Moreover we clarify a difference between the
ultrametric space and other graphs, such as cycle graph, line, hypercube and
complete graph, for the localization of the quantum case. Our quantum walk may
be useful for a quantum search algorithm on a tree-like hierarchical structure.
