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.
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