Aims. We use stellar line-of-sight velocities to constrain the dark-matter density profile of Eridanus 2, an ultra-faint dwarf galaxy with an absolute V-band magnitude MV = −7.1 that corresponds to a stellar mass M* ≈ 9 × 104 M⊙. We furthermore derive constraints on fundamental properties of self-interacting and fuzzy dark matter scenarios.Methods. We present new observations of Eridanus 2 from MUSE-Faint, a survey of ultra-faint dwarf galaxies with the Multi Unit Spectroscopic Explorer on the Very Large Telescope, and determine line-of-sight velocities for stars inside the half-light radius. Combined with literature data, we have 92 stellar tracers out to twice the half-light radius. With these tracers we constrain models of cold dark matter, self-interacting dark matter, and fuzzy dark matter, using CJAM and pyGravSphere for the dynamical analysis. The models of self-interacting and fuzzy dark matter relate the density profile to the self-interaction coefficient and the dark-matter particle mass, respectively.Results. We find substantial evidence (Bayes factor ∼10−0.6) for cold dark matter (a cuspy halo) over self-interacting dark matter (a cored halo) and weak evidence (Bayes factor ∼10−0.4) for fuzzy dark matter over cold dark matter. We find a virial mass M200 ∼ 108 M⊙ and astrophysical factors J(αcJ) ~ 1011 M⊙2 kpc−5 and D(αcD) ~ 102 − 102.5 M⊙ kpc−2 (proportional to dark-matter annihilation and decay signals, respectively), the exact values of which depend on the density profile model. The mass-to-light ratio within the half-light radius is consistent with the literature. We do not resolve a core (rc < 47 pc, 68% confidence level) or a soliton (rsol < 7.2 pc, 68% confidence level). These limits are equivalent to an effective self-interaction coefficient fΓ < 2.2 × 10−29 cm3 s−1 eV−1 c2 and a fuzzy-dark-matter particle mass ma > 4.0 × 10−20 eV c−2. The constraint on self-interaction is complementary to those from gamma-ray searches. The constraint on fuzzy-dark-matter particle mass is inconsistent with those obtained for larger dwarf galaxies, suggesting that the flattened density profiles of those galaxies are not caused by fuzzy dark matter.Key words: dark matter / galaxies: individual: Eridanus 2 / stars: kinematics and dynamics / techniques: imaging spectroscopy⋆ Based on observations made with ESO telescopes at the La Silla Paranal Observatory under programme IDs 0100.D-0807, 0102.D-0372, 0103.D-0705, and 0104.D-0199.