The nonlinear propagation of dust-ion-acoustic (DIA) waves in a dusty plasma containing trapped electrons following vortex-like distribution, cold mobile ions, and arbitrarily charged static dust is theoretically investigated. The properties of small but finite amplitude DIA solitary waves (SWs) are studied by employing the reductive perturbation technique. It is found that owing to the departure from the Maxwellian electron distribution to a vortex-like one, the dynamics of such DIA SWs is governed by a modified Korteweg-de Vries equation. The basic features (amplitude, width, speed, etc.) of such DIA SWs, which are found to be significantly modified by the vortex-like electron distribution and dust polarity, are also examined. The implications of our results to space and laboratory dusty plasmas are briefly discussed. (C) 2011 American Institute of Physics. [doi:10.1063/1.3610546]