  Given a metric space $(X,d)$, a set $S\subseteq X$ is called a
$k$-\emph{metric generator} for $X$ if any pair of different points of $X$ is
distinguished by at least $k$ elements of $S$. A $k$-\emph{metric basis} is a
$k$-metric generator of the minimum cardinality in $X$. We prove that
ultrametric spaces do not have finite $k$-metric bases for $k>2$. We also
characterize when the metric and 2-metric bases of an ultrametric space are
finite and, when they are finite, we characterize them. Finally, we prove that
an ultrametric space can be easily recovered knowing only the metric basis and
the coordinates of the points in it.
