  Aims. We use stellar line-of-sight velocities to constrain the dark
matter-density profile of Eridanus 2, an ultra-faint dwarf galaxy
($M_\mathrm{V} = -7.1$, $M_* \approx 9 \times 10^4\,M_\odot$). 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 MUSE 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. We constrain models of cold dark matter,
self-interacting dark matter, and fuzzy dark matter with these tracers, using
CJAM and pyGravSphere for the dynamical analysis. Results. We find substantial
evidence for cold dark matter over self-interacting dark matter and weak
evidence for fuzzy dark matter over cold dark matter. We find a virial mass
$M_{200} \sim 10^8\,M_\odot$ and astrophysical factors $J(\alpha_\mathrm{c}^J)
\sim 10^{11}\,M_\odot^2\,\mathrm{kpc}^{-5}$ and $D(\alpha_\mathrm{c}^D) \sim
10^2$-$10^{2.5}\,M_\odot\,\mathrm{kpc}^{-2}$. We do not resolve a core
($r_\mathrm{c} < 47\,\mathrm{pc}$, 68-% level) or soliton ($r_\mathrm{sol} <
7.2\,\mathrm{pc}$, 68-% level). These limits are equivalent to an effective
self-interaction coefficient $f\Gamma < 2.2 \times
10^{-29}\,\mathrm{cm}^3\,\mathrm{s}^{-1}\,\mathrm{eV}^{-1}\,c^2$ and a
fuzzy-dark-matter particle mass $m_\mathrm{a} > 4.0 \times
10^{-20}\,\mathrm{eV}\,c^{-2}$. 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. (Abridged)
