  Context: The 440 Myr old main-sequence A-star Fomalhaut is surrounded by an
eccentric debris belt with sharp edges. This sort of a morphology is usually
attributed to planetary perturbations, but the orbit of the only planetary
candidate detected so far, Fomalhaut b, is too eccentric to efficiently shape
the belt. Alternative models that could account for the morphology without
invoking a planet are stellar encounters and gas-dust interactions.
  Aims: We aim to test the possibility of gas-dust interactions as the origin
of the observed morphology by putting upper limits on the total gas content of
the Fomalhaut belt.
  Methods: We derive upper limits on the CII 158 $\mu$m and OI 63 $\mu$m
emission by using non-detections from the Photodetector Array Camera and
Spectrometer (PACS) onboard the Herschel Space Observatory. Line fluxes are
converted into total gas mass using the non-local thermodynamic equilibrium
(non-LTE) code RADEX. We consider two different cases for the elemental
abundances of the gas: solar abundances and abundances similar to those
observed for the gas in the $\beta$ Pictoris debris disc.
  Results: The gas mass is shown to be below the millimetre dust mass by a
factor of at least $\sim$3 (for solar abundances) respectively $\sim$300 (for
$\beta$ Pic-like abundances).
  Conclusions: The lack of gas co-spatial with the dust implies that gas-dust
interactions cannot efficiently shape the Fomalhaut debris belt. The morphology
is therefore more likely due to a yet unseen planet (Fomalhaut c) or stellar
encounters.
