We measured low-temperature electrical resistivity in a magnetic field as a function of pressure, and studied the superconducting property of CeIrSi3 with a non-centrosymmetric tetragonal structure. CeIrSi3 is an antiferromagnet with a Neel temperature T-N = 5.0 K at ambient pressure, but the antiferromagnetic ordering disappears completely at a pressure P-c = 2.25 GPa. Superconductivity appears in a wide pressure region from 1.8 to about 3.5 GPa, with a maximum transition temperature T-sc similar or equal to 1.6 K at P-c* similar or equal to 2.63 GPa. At around P-c*, the upper critical field H-c2 for the magnetic field H along the [0 01] direction is not destroyed by spin polarization based on Zeeman coupling but possesses an upturn curvature below 1K, revealing a divergent tendency of H-c2 at P-c*, with a huge H-c2(0) similar or equal to 450 kOe. On the other hand, the upper critical field for H parallel to [110] indicates the tendency of Pauli paramagnetic suppression, with H-c2(0) = 95 kOe at 2.65 GPa. This might be a combined phenomenon between the electronic instability at P-c* and the characteristic superconducting property without inversion symmetry in the tetragonal structure.