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The basic features of nonlinear ion acoustic (IA) waves are theoretically studied in a superthermal electron–positron–ion (e–p–i) plasma with weakly transverse perturbation. A three-dimensional Kadomtsev–Petviashvili (KP) equation governing evolution of weakly nonlinear IA waves is derived by means of a reductive perturbation method. The energy integral equation is used to study the existence domain of the localized structures. It is found that deviation from thermodynamics equilibrium increases the existence domain of solitary solution and also makes the IA solitary structure more spiky. The ion concentration has an important effect on the existence domain of solitary solution, as for low ion density the primitive domain reduces significantly.