The two-dimensional dynamics of nonlinear ion-acoustic waves in a weakly magnetized plasma comprising cold ions and trapped as well as free electrons is considered. It is shown that owing to departure from the Boltzmann electron distribution to a vortex-like one, the dynamics of small but finite-amplitude ion-acoustic waves is governed by a new nonlinear equation which is valid for both unmagnetized and magnetized plasmas. For exactly vanishing magnetic fields the modified Kadomtsev–Petviashvili (mKP) equation is recovered. For weak magnetic fields, however, the dynamics is mainly different from the mKP equation, depending on the amplitude. By increasing the magnetic field, the new equation becomes similar (but not identical) to the modified Zakharov–Kuznetsov (mZK) equation, which is fulfilled for very strong magnetic fields. The plane periodic and solitary wave solutions of this equation are obtained using the appropriate scalings.