  We use the Keck Cosmic Web Imager integral-field unit spectrograph to: 1)
measure the global stellar population parameters for the ultra-diffuse galaxy
(UDG) Dragonfly 44 (DF44) to much higher precision than previously possible for
any UDG, and 2) for the first time measure spatially-resolved stellar
population parameters of a UDG. We find that DF44 falls below the
mass--metallicity relation established by canonical dwarf galaxies both in and
beyond the Local Group. We measure a flat radial age gradient ($m_{\rm age}
\sim +0.01_{-0.08}^{+0.07}$ log Gyr kpc$^{-1}$) and a flat-to-positive
metallicity gradient ($m_{\rm [Fe/H]} \sim +0.08_{-0.11}^{+0.11}$ dex
kpc$^{-1}$), which are inconsistent with the gradients measured in similarly
pressure-supported dwarf galaxies. We also measure a flat-to-negative [Mg/Fe]
gradient ($m_{\rm [Mg/Fe]} \sim -0.18_{-0.17}^{+0.17}$ dex kpc$^{-1}$) such
that the central $1.5$ kpc of DF44 has stellar population parameters comparable
to metal-poor globular clusters. Overall, DF44 does not have internal
properties similar to other dwarf galaxies and is inconsistent with it having
been puffed up through a prolonged, bursty star-formation history, as suggested
by some simulations. Rather, the evidence indicates that DF44 experienced an
intense epoch of "inside-out" star formation and then quenched early and
catastrophically, such that star-formation was cut off more quickly than in
canonical dwarf galaxies.
