We present the first comprehensive archival study of the X-ray properties of ultracompact dwarf (UCD) galaxies, with the goal of identifying weakly-accreting central black holes in UCDs. Our study spans 578 UCDs distributed across thirteen different host systems, including clusters, groups, fossil groups, and isolated galaxies. Of the 336 spectroscopically-confirmed UCDs with usable archival Chandra imaging observations, 21 are X-ray-detected. Imposing a completeness limit of $L_X>2times10^{38}$ erg s$^{-1}$, the global X-ray detection fraction for the UCD population is $sim3%$. Of the 21 X-ray-detected UCDs, seven show evidence of long-term X-ray time variability on the order of months to years. X-ray-detected UCDs tend to be more compact than non-X-ray-detected UCDs, and we find tentative evidence that the X-ray detection fraction increases with surface luminosity density and global stellar velocity dispersion. The X-ray emission of UCDs is fully consistent with arising from a population of low-mass X-ray binaries (LMXBs). In fact, there are fewer X-ray sources than expected using a naive extrapolation from globular clusters. Invoking the fundamental plane of black hole activity for SUCD1 near the Sombrero galaxy, for which archival Jansky Very Large Array imaging at 5 GHz is publicly available, we set an upper limit on the mass of a hypothetical central black hole in that UCD to be $lesssim10^5M_{odot}$. While the majority of our sources are likely LMXBs, we cannot rule out central black holes in some UCDs based on X-rays alone, and so we address the utility of follow-up radio observations to find weakly-accreting central black holes.
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