This work presents the observations and analysis of ATLAS19dqr/SN 2019bkc, an extraordinary rapidly evolving transient event located in an isolated environment, tens of kiloparsecs from any likely host. Its light curves rise to maximum light in 5-6 d and then display a decline of Δm15 ∼ 5 mag. With such a pronounced decay, it has one of the most rapidly evolving light curves known for a stellar explosion. The early spectra show similarities to normal and "ultra-stripped" type Ic SNe, but the early nebular phase spectra, which were reached just over two weeks after explosion, display prominent calcium lines, marking SN 2019bkc as a Ca-rich transient. The Ca emission lines at this phase show an unprecedented and unexplained blueshift of 10 000-12 000 km s-1. Modelling of the light curve and the early spectra suggests that the transient had a low ejecta mass of 0.2-0.4 M⊙ and a low kinetic energy of (2-4) × 1050 erg, giving a specific kinetic energy Ek/Mej ∼ 1 [1051 erg]/M⊙. The origin of this event cannot be unambiguously defined. While the abundance distribution used to model the spectra marginally favours a progenitor of white dwarf origin through the tentative identification of Ar II, the specific kinetic energy, which is defined by the explosion mechanism, is found to be more similar to an ultra-stripped core-collapse events. SN 2019bkc adds to the diverse range of physical properties shown by Ca-rich events.Partially based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programmes 199.D-0143 and 0102.D-0137(A).