A subset $A$ of a uniform space is said to be bounded if for each entourage $V$, $A$ is a subset of $V^n[F]$ for some natural number $n$ and some finite set $F$. A subset of a metric space is said to be bounded if it is contained in some open ball. Now if $U$ is the uniformity induced by a metric $d$, then the open balls with respect to $d$ are entourages in $U$, so clearly a set bounded with respect to $d$ is also bounded with respect to $U$.
But this journal paper says that the converse is not true:
In a metric space $(X,d)$ we have that each set that is bounded for the metric $d$ is bounded ... for the underlying uniformity, but the converse is in general not true.
So my question is, what is an example of a metric space $(X,d)$ where some sets bounded with respect to the the uniformity induced by $d$ are not bounded with respect to $d$?