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We consider the group $SL_2(K)$, where $K$ is a local non-archimedean fieldof characteristic two. We prove that the depth of any irreduciblerepresentation of $SL_2 (K)$ is larger than the depth of the correspondingLanglands parameter, with equality if and only if the L-parameter isessentially tame.  We also work out a classification of all $L$-packets for $SL_2 (K)$ and forits non-split inner form, and we provide explicit formulae for the depths oftheir $L$-parameters.

On $L$-packets and depth for $SL_2(K)$ and its inner form