  Rotation was shown to have a strong impact on the structure and light element
nucleosynthesis in massive stars. In particular, models including rotation can
reproduce the primary nitrogen observed in halo extremely metal-poor (EMP)
stars. Additional exploratory models showed that rotation may enhance
$s$-process production at low metallicity.
  Here we present a large grid of massive star models including rotation and a
full $s$-process network to study the impact of rotation on the weak
$s$-process. We explore the possibility of producing significant amounts of
elements beyond the strontium peak, which is where the weak $s$-process usually
stops.
  We used the Geneva stellar evolution code coupled to an enlarged reaction
network with 737 nuclear species up to bismuth to calculate
$15-40\,\text{M}_\odot$ models at four metallicities ($Z = 0.014,10^{-3}$,
$10^{-5}$, and $10^{-7}$) from the main sequence up to the end of oxygen
burning.
  We confirm that rotation-induced mixing between the convective H-shell and
He-core enables an important production of primary $^{14}$N and $^{22}$Ne and
$s$-process at low metallicity. At low metallicity, even though the production
is still limited by the initial number of iron seeds, rotation enhances the
$s$-process production, even for isotopes heavier than strontium, by increasing
the neutron to seed ratio. The increase in this ratio is a direct consequence
of the primary production of $^{22}$Ne. Despite nuclear uncertainties affecting
the $s$-process production and stellar uncertainties affecting the
rotation-induced mixing, our results show a robust production of $s$ process at
low metallicity when rotation is taken into account. Considering models with a
distribution of initial rotation rates enables to reproduce the observed large
range of the [Sr/Ba] ratios in (carbon-enhanced and normal) EMP stars.
