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Rotation was shown to have a strong impact on the structure and light elementnucleosynthesis in massive stars. In particular, models including rotation canreproduce 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 afull $s$-process network to study the impact of rotation on the weak$s$-process. We explore the possibility of producing significant amounts ofelements beyond the strontium peak, which is where the weak $s$-process usuallystops.  We used the Geneva stellar evolution code coupled to an enlarged reactionnetwork 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 oxygenburning.  We confirm that rotation-induced mixing between the convective H-shell andHe-core enables an important production of primary $^{14}$N and $^{22}$Ne and$s$-process at low metallicity. At low metallicity, even though the productionis still limited by the initial number of iron seeds, rotation enhances the$s$-process production, even for isotopes heavier than strontium, by increasingthe neutron to seed ratio. The increase in this ratio is a direct consequenceof the primary production of $^{22}$Ne. Despite nuclear uncertainties affectingthe $s$-process production and stellar uncertainties affecting therotation-induced mixing, our results show a robust production of $s$ process atlow metallicity when rotation is taken into account. Considering models with adistribution of initial rotation rates enables to reproduce the observed largerange of the [Sr/Ba] ratios in (carbon-enhanced and normal) EMP stars.

$s$-process production in rotating massive stars at solar and low metallicities