We report observations of the hydrogen- and helium-deficient supernova (SN) 2019bkc/ATLAS19dqr. With B- and r-band decline between peak and 10 days post peak of Delta m_10(B) = 4.9+/-0.1 mag and Delta m_10(r) = 4.1+/-0.1 mag, respectively, SN 2019bkc is the most rapidly-declining Type I SN discovered so far. The light curves and spectra of SN 2019bkc show some unprecedented characteristics compared to known SNe, while its closest matches are the rapidly-declining SN 2005ek and SN 2010X. 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. SN 2019bkc poses a challenge for existing models of fast-evolving SNe.