  We explore the kinematic gas properties of six $5.5<z<7.4$ galaxies in the
JWST Advanced Deep Extragalactic Survey (JADES), using high-resolution
JWST/NIRSpec multi-object spectroscopy of the rest-frame optical emission lines
[OIII] and H$\alpha$. The objects are small and of low stellar mass ($\sim
1\,$kpc; $M_*\sim10^{7-9}\,{\rm M_\odot}$), less massive than any galaxy
studied kinematically at $z>1$ thus far. The cold gas masses implied by the
observed star formation rates are $\sim 10\times$ larger than the stellar
masses. We find that their ionised gas is spatially resolved by JWST, with
evidence for broadened lines and spatial velocity gradients. Using a simple
thin-disc model, we fit these data with a novel forward modelling software that
accounts for the complex geometry, point spread function, and pixellation of
the NIRSpec instrument. We find the sample to include both rotation- and
dispersion-dominated structures, as we detect velocity gradients of $v(r_{\rm
e})\approx100-150\,{\rm km\,s^{-1}}$, and find velocity dispersions of
$\sigma_0\approx 30-70\,{\rm km\,s^{-1}}$ that are comparable to those at
cosmic noon. The dynamical masses implied by these models ($M_{\rm
dyn}\sim10^{9-10}\,{\rm M_\odot}$) are larger than the stellar masses by up to
a factor 40, and larger than the total baryonic mass (gas + stars) by a factor
of $\sim 3$. Qualitatively, this result is robust even if the observed velocity
gradients reflect ongoing mergers rather than rotating discs. Unless the
observed emission line kinematics is dominated by outflows, this implies that
the centres of these galaxies are dark-matter dominated or that star formation
is $3\times$ less efficient, leading to higher inferred gas masses.
