  Let $p$ be a prime number and $(K, v)$ a Henselian valued field with a
residue field $\widehat K$. This paper determines the Brauer $p$-dimension of
$K$, in case $p \neq {\rm char}(\widehat K)$ and $\widehat K$ is a
$p$-quasilocal field properly included in its maximal $p$-extension. When
$\widehat K$ is a local field, it describes index-exponent pairs of central
division $K$-algebras of $p$-primary degrees. The same goal is achieved, if
$(K, v)$ is maximally complete, char$(K) = p$ and $\widehat K$ is local.
