  Let $A$ be a finite subset of $\ffield$, the field of Laurent series in $1/t$
over a finite field $\mathbb{F}_q$. We show that for any $\epsilon>0$ there
exists a constant $C$ dependent only on $\epsilon$ and $q$ such that
$\max\{|A+A|,|AA|\}\geq C |A|^{6/5-\epsilon}$. In particular such a result is
obtained for the rational function field $\mathbb{F}_q(t)$. Identical results
are also obtained for finite subsets of the $p$-adic field $\mathbb{Q}_p$ for
any prime $p$.
