Both linear and weakly nonlinear dust‐acoustic (DA) solitons propagation are revisited in the presence of adiabatically trapped‐nonextensive ions. A physically relevant distribution (called Tsallis‐Gurevich ions distribution) is outlined here for the first time. The effect of particle trapping has been taken into account, resulting in a new expression for the ion density. The role a background ion nonextensivity may play on the main proprieties (viz., dispersion relation and soliton's profile) of DA mode, inherent to space dusty plasma, is then analysed. In the q > 1 case, we have shown that as the nonextensive character of trapped ions increases in the plasma, the potential pulse amplitude increases while its width is narrowed. The modifications prompted by the presence of adiabatically trapped‐nonextensive ions on both DA energy and solitary wave's electric field are also analysed. Interestingly, we have found that DA soliton energy increases as the ions evolve far away from their Maxwellian extensive trapping. Also, it is found that, for q > 1, the stronger the inter‐ions correlation, the stronger the electric field. Our investigation, motivated by space and laboratory plasma observations of plasmas containing non‐Maxwellian particles alongside trapped particles, may complement and provide new insight into previously published works dealing with solitary waves in plasma.