The first extragalactic ultra-compact X-ray binary : a candidate black hole-white dwarf system

M31 UCXB-1 is one of the brightest X-ray point sources in the bulge of M31, with a peak X-ray luminosity $L_{\mathrm{0.5-10 \: keV}} = 2.9^{+0.2}_{-0.2} \times 10^{38} \: \mathrm{erg} \: \mathrm{s}^{-1}$. Both XMM-Newton and Chandra observations have detected an eclipsing signal with a period of about 465 seconds from this source, and we note that the periodic signal is detected exclusively during the source's high-luminosity states. This signal probably originates from its orbital motion, therefore it is an ultra-compact X-ray binary (UCXB) candidate with the highest X-ray luminosity. Our theoretical analyses show that M31 UCXB-1 is in good agreement with the luminosity-orbital period relation ($L_{\mathrm{2-10 \: keV}}-P_{\mathrm{orb}}$) of the black hole/neutron star--white dwarf (BH/NS--WD) UCXB system. Moreover, our spectral analyses indicate that the primary in M31 UCXB-1 is more likely to be a BH rather than an NS. The results show that M31 UCXB-1 is a BH--WD system, with the shortest orbital period, the possibly strongest gravitational wave emission, and the most massive white dwarf among the known UCXBs.