  Pressure induced superconductivity in non-centrosymmetric CeRhSi$_{3}$ and
CeIrSi$_{3}$ compounds has attracted significant attention of the scientific
community since its discovery 15 years ago. Up-to-date, all reported
experimental results were obtained employing the hybrid-cylinder piston
pressure cells with a maximum reachable pressure of 3 GPa. Present study
focuses on the superconducting state at higher, so far unreported, pressures
using the Bridgman anvil cell and a CeRhSi$_{3}$ single crystal synthesized by
the Sn-true-flux method. The initial increase of superconducting critical
temperature from 0.4 K at 1.1 GPa to 1.1 K at 2.4 GPa is followed by a gradual
suppression of SC state upon increasing the pressure above 3.0 GPa, forming a
typical dome. The pressure induced superconductivity is expected to be
completely suppressed in the pressure region between 4.5 and 5.0 GPa.
Temperature dependence of electrical resistivity in constant magnetic fields
and high pressures, as well as the magnetoresistance measurements, reveal a
large critical field, exceeding 19 T at 0.6 K and 2.4 GPa, sharply decreasing
receding the superconductivity dome. The previously reported $\it{T-p}$ and
$\it{H-T}$ phase diagrams are completed by our high-pressure data and discussed
in the frame of previous results.
