Three independent nonperturbative calculations are reported for the electron-impact ionization of both the ground and first excited states of the neutral lithium atom. The time-dependent close-coupling, the R matrix with pseudostates, and the converged close-coupling methods yield total integral cross sections that are in very good agreement with each other, while perturbative distorted-wave calculations yield cross sections that are substantially higher. These nonperturbative calculations provide a benchmark for the continued development of electron-atom experimental methods designed to measure both ground and excited state ionization.