Graev ultrametrics and surjectively universal non-Archimedean Polish groups

We define Graev-type ultrametrics and use them to show the existence of surjectively universal non-Archimedean Polish groups. We also show the existence of surjectively universal abelian non-Archimedean Polish groups and surjectively universal non-Archimedean Polish groups admitting compatible two-sided invariant metrics.