We present a pressure study of the electrical resistivity, AC magnetic susceptibility and powder x-ray diffraction (XRD) of the newly discovered BiS$_2$-based superconductor EuBiS$_2$F. At ambient pressure, EuBiS$_2$F shows an anomaly in the resistivity at around $T_0approx 280$ K and a superconducting transition at $T_capprox 0.3$ K. Upon applying hydrostatic pressure, there is little change in $T_0$ but the amplitude of the resistive anomaly is suppressed, whereas there is a dramatic enhancement of $T_c$ from 0.3 K to about 8.6 K at a critical pressure of $p_c$ $approx{1.4}$ GPa. XRD measurements confirm that this enhancement of $T_c$ coincides with a structural phase transition from a tetragonal phase ($P4/nmm$) to a monoclinic phase ($P2_1$/m), which is similar to that observed in isostructural LaO$_{0.5}$F$_{0.5}$BiS$_2$. Our results suggest the presence of two different superconducting phases with distinct crystal structures in EuBiS$_2$F, which may be a general property of this family of BiS$_2$-based superconductors.
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