  We present spatially-resolved stellar kinematics of the well-studied ultra
diffuse galaxy (UDG) Dragonfly 44, as determined from 25.3 hrs of observations
with the Keck Cosmic Web Imager. The luminosity-weighted dispersion within the
half-light radius is $\sigma_{1/2}=33^{+3}_{-3}$ km/s. There is no evidence for
rotation, with $V/\sigma<0.12$ (90% confidence) along the major axis, in
apparent conflict with models where UDGs are the high-spin tail of the normal
dwarf galaxy distribution. The spatially-averaged line profile is more peaked
than a Gaussian, with Gauss-Hermite coefficient $h_4=0.13\pm 0.05$. The
mass-to-light ratio within the effective radius is $M/L=26^{+7}_{-6}$, similar
to other UDGs and higher by a factor of six than normal galaxies of the same
luminosity. This difference between UDGs and other galaxies is, however,
sensitive to the aperture that is used, and is much reduced when the $M/L$
ratios are measured within a fixed radius of 10 kpc. Dragonfly 44 has a rising
velocity dispersion profile, from $\sigma=26^{+4}_{-4}$ km/s at R=0.2 kpc to
$\sigma=41^{+8}_{-8}$ km/s at R=5.1 kpc. The profile can only be fit with a
cuspy NFW profile if the orbital distribution has strong tangential anisotropy,
with $\beta=-0.8^{+0.4}_{-0.5}$. An alternative explanation is that the dark
matter profile has a core: a Di Cintio et al. (2014) density profile with a
mass-dependent core provides a very good fit to the kinematics for a halo mass
of $\log (M_{200}/{\rm M}_{\odot})=11.2^{+0.6}_{-0.6}$ and
$\beta=-0.1^{+0.2}_{-0.3}$, i.e., isotropic orbits. This model predicts a
slight positive kurtosis, in qualitative agreement with the measured $h_4$
parameter. UDGs such as Dragonfly 44 are dark matter dominated even in their
centers, and can constrain the properties of dark matter in a regime where
baryons usually dominate the kinematics: small spatial scales in massive halos.
