  Let $(K, v)$ be a Henselian discrete valued field with residue field
$\widehat K$ of characteristic $q \ge 0$, and Brd$_{p}(K)$ be the Brauer
$p$-dimension of $K$, for each prime $p$. The present paper shows that if $p =
q$, then Brd$_{p}(K) \le 1$ if and only if $\widehat K$ is a $p$-quasilocal
field and the degree $[\widehat K\colon \widehat K ^{p}]$ is $\le p$. This
complements our earlier result that, in case $p \neq q$, we have Brd$_{p}(K)
\le 1$ if and only if $\widehat K$ is $p$-quasilocal and Brd$_{p}(\widehat K)
\le 1$.
