  [Abridged] Aims. The lowest-mass galaxies, ultra-faint dwarf galaxies,
promise unparalleled constraints on how feedback regulates galaxy formation,
and on the small-scale matter power spectrum. Their inner dark-matter densities
can also be used to constrain dark-matter models. In this paper, we present 201
new stellar line-of-sight velocities from the MUSE-Faint survey for the
(ultra-)faint dwarf galaxies Antlia B, Leo T, Hydra II, and Grus 1. Combining
these with literature data, we obtain the tightest constraints to date on their
dark-matter halo masses and inner dark-matter densities. Methods. We use the
Jeans equations implemented in CJAM to model the density profiles and constrain
the presence of dark-matter cores and solitons (a prediction of fuzzy
dark-matter models). Further modelling is done with GravSphere to test the
influence of the choice of modelling tool. We calculate masses, concentrations,
and circular velocities from the profiles, include results for Eridanus 2 from
our previous work, and compare these properties to theoretical scaling
relations, deriving constraints on tidal stripping in the process. Results. We
find that dark-matter cores as large as those of more massive dwarf galaxies
are ruled out for our galaxies (core radius $r_\mathrm{c} <
66$-$95\,\mathrm{pc}$ at the 68% confidence level). We constrain the soliton
radii to $r_\mathrm{sol} < 13$-$112\,\mathrm{pc}$ (68% confidence level). We
find that the galaxies are consistent with not having been significantly
tidally stripped within their half-light radii. The virial masses and
concentrations are sensitive to the choice of dynamical modelling tool:
GravSphere produces results consistent with $M_{200} \sim 10^9\,M_\odot$, as
expected from models in which ultra-faint dwarf galaxies are re-ionization
fossils, while CJAM prefers haloes that are less massive.
