  It is a long-standing open question whether every Polish group that is not
locally compact admits a Borel action on a standard Borel space whose
associated orbit equivalence relation is not essentially countable. We answer
this question positively for the class of all Polish groups that embed in the
isometry group of a locally compact metric space. This class contains all
non-archimedean Polish groups, for which we provide an alternative proof based
on a new criterion for non-essential countability. Finally, we provide the
following variant of a theorem of Solecki: every infinite-dimensional Banach
space has a continuous action whose orbit equivalence relation is Borel but not
essentially countable.
