  Let $n>1$ be a positive integer. Let $R$ be a henselian local ring with
residue field $k$ of $n$th level $s_n(k)$. We give some upper and lower bounds
for the $n$th Waring number $w_n(R)$ in terms of $w_n(k)$ and $s_n(k)$. In
large number of cases we are able to compute $w_n(R)$. Similar results for the
$n$th Waring number of the total ring of fractions of $R$ are obtained. We then
provide applications. In particular we compute $w_n(\mathbb{Z}_p)$ and
$w_n(\mathbb{Q}_p)$ for $n\in\{3,4,5\}$ and any prime $p$.
