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We investigate the high-pressure phase of the iron-based superconductorFeSe$_{0.89}$S$_{0.11}$ using transport and tunnel diode oscillator studies. Weconstruct detailed pressure-temperature phase diagrams that indicate thatoutside of the nematic phase, the superconducting critical temperature reachesa minimum before it is quickly enhanced towards 40 K above 4 GPa. Theresistivity data reveal signatures of a fan-like structure of non-Fermi liquidbehaviour which could indicate the existence of a putative quantum criticalpoint buried underneath the superconducting dome around 4.3 GPa. Furtherincreasing the pressure, the zero-field electrical resistivity develops anon-metallic temperature dependence and the superconducting transition broadenssignificantly. Eventually, the system fails to reach a fully zero-resistancestate despite a continuous finite superconducting transition temperature, andany remaining resistance at low temperatures becomes stronglycurrent-dependent. Our results suggest that the high-pressure, high-$T_c$ phaseof iron chalcogenides is very fragile and sensitive to uniaxial effects of thepressure medium, cell design and sample thickness which can trigger afirst-order transition. These high-pressure regions could be understoodassuming a real-space phase separation caused by concomitant electronic andstructural instabilities.

Collapse of Metallicity and High-$T_c$ Superconductivity in the High-Pressure phase of FeSe$_{0.89}$S$_{0.11}$