  The standard period-index conjecture for the Brauer group of a field of
transcendence degree 2 over a $p$-adic field predicts that the index divides
the cube of the period. Using Gabber's theory of prime-to-$\ell$ alterations
and the deformation theory of twisted sheaves, we prove that the index divides
the fourth power of the period for every Brauer class whose period is prime to
$6p$, giving the first uniform period-index bounds over such fields.
