We report observations of the hydrogen-deficient supernova (SN) 2019bkc/ATLAS19dqr. With B- and r-band decline between peak and 10 days post peak of Delta m_10(B)=5.24+/-0.07 mag and Delta m_10(r)=3.85+/-0.10$ mag, respectively, SN 2019bkc is the most rapidly declining SN I discovered so far. While its closest matches are the rapidly declining SN 2005ek and SN 2010X, the light curves and spectra of SN 2019bkc show some unprecedented characteristics. SN 2019bkc appears "hostless," with no identifiable host galaxy near its location, although it may be associated with the galaxy cluster MKW1 at z = 0.02. We evaluate a number of existing models of fast-evolving SNe, and we find that none of them can satisfactorily explain all aspects of SN 2019bkc observations.