We report on a multi-mJ 2.1 μm optical parametric chirped-pulse amplification (OPCPA) system operating at 1 kHz repetition rate, pumped by a picosecond cryogenic Yb:YAG laser, and the demonstration of soft x-ray high-harmonic generation (HHG) with a flux of ∼2×10(8) photon/s/1% bandwidth at 160 eV in Ar. The 1 kHz cryogenic Yb:YAG pump laser amplifies pulses up to 56 mJ and delivers compressed 42 mJ, 17 ps pulses to the 2.1 μm OPCPA system. In the three-stage OPCPA chain, we have obtained up to 2.6 mJ of output energies at 2.1 μm and pulses compressed to 40 fs with good beam quality. Finally, we show cut-off extension of HHG driven by this 2.1 μm source in Ar and N2 gas cells to 190 eV with high photon flux. Our 3D propagation simulation confirms the generation of soft x-ray attosecond pulses from the experiment with Ar.