On a characterization theorem on non-discrete totally disconnected locally compact fields

We prove the following theorem. Let $X$ be a non-discrete totally disconnected locally compact field, $R$ be its ring of integers, $P$ be the nonzero prime ideal of $R$. Assume that the residue field $R/P$ is a field of characteristic $p>2$. Let $\xi$ and $\eta$ be independent identically distributed random variables with values in $X$ and distribution $\mu$, such that $\mu$ has a continuous density with respect to a Haar measure on $X$. This implies that the random variables $S=\xi+\eta$ and $D=(\xi-\eta)^2$ are independent if and only if $\mu$ is a shift of the Haar distribution of a compact subgroup of $X$.