  We consider the group $SL_2(K)$, where $K$ is a local non-archimedean field
of characteristic two. We prove that the depth of any irreducible
representation of $SL_2 (K)$ is larger than the depth of the corresponding
Langlands parameter, with equality if and only if the L-parameter is
essentially tame.
  We also work out a classification of all $L$-packets for $SL_2 (K)$ and for
its non-split inner form, and we provide explicit formulae for the depths of
their $L$-parameters.
