  Let $K$ be a non-archimedean local field. We show that discrete subgroups
without 2-torsion in $\mathrm{PSL}_2(K)$ can always be lifted to
$\mathrm{SL}_2(K)$, and provide examples (when $\mathrm{char}(K) \neq 2$) which
cannot be lifted if either of these conditions is removed. We also briefly
discuss lifting representations of groups into $\mathrm{PSL}_2(K)$ to
$\mathrm{SL}_2(K)$.
