  We investigate the high-pressure phase of the iron-based superconductor
FeSe$_{0.89}$S$_{0.11}$ using transport and tunnel diode oscillator studies. We
construct detailed pressure-temperature phase diagrams that indicate that
outside of the nematic phase, the superconducting critical temperature reaches
a minimum before it is quickly enhanced towards 40 K above 4 GPa. The
resistivity data reveal signatures of a fan-like structure of non-Fermi liquid
behaviour which could indicate the existence of a putative quantum critical
point buried underneath the superconducting dome around 4.3 GPa. Further
increasing the pressure, the zero-field electrical resistivity develops a
non-metallic temperature dependence and the superconducting transition broadens
significantly. Eventually, the system fails to reach a fully zero-resistance
state despite a continuous finite superconducting transition temperature, and
any remaining resistance at low temperatures becomes strongly
current-dependent. Our results suggest that the high-pressure, high-$T_c$ phase
of iron chalcogenides is very fragile and sensitive to uniaxial effects of the
pressure medium, cell design and sample thickness which can trigger a
first-order transition. These high-pressure regions could be understood
assuming a real-space phase separation caused by concomitant electronic and
structural instabilities.
