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[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 densitiescan also be used to constrain dark-matter models. In this paper, we present 201new 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. Combiningthese with literature data, we obtain the tightest constraints to date on theirdark-matter halo masses and inner dark-matter densities. Methods. We use theJeans equations implemented in CJAM to model the density profiles and constrainthe presence of dark-matter cores and solitons (a prediction of fuzzydark-matter models). Further modelling is done with GravSphere to test theinfluence of the choice of modelling tool. We calculate masses, concentrations,and circular velocities from the profiles, include results for Eridanus 2 fromour previous work, and compare these properties to theoretical scalingrelations, deriving constraints on tidal stripping in the process. Results. Wefind that dark-matter cores as large as those of more massive dwarf galaxiesare ruled out for our galaxies (core radius $r_\mathrm{c} <66$-$95\,\mathrm{pc}$ at the 68% confidence level). We constrain the solitonradii to $r_\mathrm{sol} < 13$-$112\,\mathrm{pc}$ (68% confidence level). Wefind that the galaxies are consistent with not having been significantlytidally stripped within their half-light radii. The virial masses andconcentrations are sensitive to the choice of dynamical modelling tool:GravSphere produces results consistent with $M_{200} \sim 10^9\,M_\odot$, asexpected from models in which ultra-faint dwarf galaxies are re-ionizationfossils, while CJAM prefers haloes that are less massive.

The MUSE-Faint survey. III. No large dark-matter cores and no significant tidal stripping in ultra-faint dwarf galaxies