A two‐dimensional study is presented, based on computer simulation, of strong electron‐ion streaming instabilities induced by counterstreaming ion beams, for initially cold electrons(ve ≪ vd).Wave growth initially agrees well with linear theory. Higher harmonics appear later, in good agreement with mode coupling theory. The wave spectrum, initially very broad perpendicular to the streaming direction, narrows as the electrons heat and the waves saturate, in accordance with quasilinear theory. Electron heating is explained by nonresonant quasilinear diffusion at early times, and by resonant turbulent diffusion at later times. In contrast to previous one‐dimensional studies, it is found that electron heating continues well after wave saturation and afterve > vd.However, the heating rate at late times is somewhat reduced by partial plateau formation in the electron velocity distribution. Coherent electron trapping is not seen, as it is in the one‐dimensional situation, but coherent ion trapping does occur and dominates the wave dynamics at late times, terminating wave growth on the time scale studied.