Let $U$ be a unipotent upper triangluar group over a local field $K$ of characteristic zero. Can we guarantee that there is a right translation invariant metric on $U$ such that any ball of finite radius is relatively compact?
Let $G$ be a locally compact group. If $G$ is compactly generated, then word length with respect to a compact generating subset defines an invariant metric which is proper (i.e. closed balls are compact). The problem here is that a unipotent group $U$ is usually not compactly generated (if $K$ is non-archimedean). But it can be naturally embedded as a closed subgroup in a compactly generated group, e.g. the subgroup $B$ of all triangular matrices. So take the word metric in $B$ and restrict to $U$.