In Mott insulators, the strong electron-electron Coulomb repulsion localizes electrons. In dimensions greater than one, their spins are usually ordered antiferromagnetically at low temperatures. Geometrical frustrations can destroy this long-range order, leading to exotic quantum spin liquid states. However, their magnetic ground states have been a long-standing mystery. Here we show that a quantum spin liquid state in the organic Mott insulator EtMe(3)Sb[Pd(dmit)(2)](2) (where Et is C(2)H(5)-, Me is CH(3)-, and dmit is 1,3-dithiole-2-thione-4,5-dithiolate) with two-dimensional triangular lattice has Pauli-paramagnetic-like low-energy excitations, which are a hallmark of itinerant fermions. Our torque magnetometry down to low temperatures (30 mK) up to high fields (32 T) reveals distinct residual paramagnetic susceptibility comparable to that in a half-filled two-dimensional metal, demonstrating the magnetically gapless nature of the ground state. Moreover, our results are robust against deuteration, pointing toward the emergence of an extended 'quantum critical phase', in which low-energy spin excitations behave as in paramagnetic metals with Fermi surface, despite the frozen charge degree of freedom.