We study particle acceleration in strongly turbulent pair plasmas using novel 3D particle-in-cell simulations, featuring particle injection from an external heat bath and diffusive escape. We demonstrate the formation of steady-state, nonthermal particle distributions with maximum energies reaching the Hillas limit. The steady state is characterized by the equilibration of plasma kinetic and magnetic pressures, which imposes upper limits on the acceleration rate. With growing cold plasma magnetization 𝜎0, nonthermal power-law spectra become harder, and the fraction of energy channeled into escaping cosmic rays increases. At 𝜎0 ≳1, the escaping cosmic rays amount to more than 50% of the dissipated energy. Our method allows for kinetic studies of particle acceleration under steady-state conditions, with applications to a variety of astrophysical systems.