  We introduce the use of $p$-descent techniques for elliptic surfaces over a
perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we
obtain an upper bound for the rank of a non-constant elliptic surface. When
$p=2$, this bound is an arithmetic refinement of a well-known geometric bound
for the rank deduced from Igusa's inequality. This answers a question raised by
Ulmer. We give some applications to rank bounds for elliptic surfaces over the
rational numbers.
