The iron-based superconductor Ba$_{1-x}$K$_x$Fe$_text{2}$As$_text{2}$ is emerging as a key material for high magnetic field applications owing to the recent developments in superconducting wires and bulk permanent magnets. Epitaxial thin films play important roles in investigating and artificially tuning physical properties; nevertheless, the synthesis of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ epitaxial thin films remained challenging because of the high volatility of K. Herein, we report the successful growth of epitaxial Ba$_{1-x}$K$_x$Fe$_text{2}$As$_text{2}$ thin films by molecular-beam epitaxy with employing a combination of fluoride substrates (CaF$_text{2}$, SrF$_text{2}$, and BaF$_text{2}$) and a low growth temperature (350$-$420$^circ$C). Our epitaxial thin film grown on CaF$_text{2}$ showed sharp superconducting transition at an onset critical temperature of 36 K, slightly lower than bulk crystals by ~2 K due presumably to the strain effect arising from the lattice and thermal expansion mismatch. Critical current density ($J$$_text{c}$) determined by the magnetization hysteresis loop is as high as 2.2 MA/cm$^text{2}$ at 4 K under self-field. In-field $J$$_text{c}$ characteristics of the film are superior to the bulk crystals. The realization of epitaxial thin films opens opportunities for tuning superconducting properties by epitaxial strain and revealing intrinsic grain boundary transport of Ba$_{1-x}$K$_x$Fe$_text{2}$As$_text{2}$.
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