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The binding energy of an off-center hydrogen-like impurity in an ultra-wide band gap β-Ga₂O₃/(Al<i>_x</i>Ga_1−<i>x</i>)₂O₃ core/shell nanostructure is studied using a variational method combined with a finite-difference algorithm. Four impurity states with the radial and axial quantum numbers being 0 or 1 in two kinds of core/shell nanostructures, including nanorods and double-walled nanotubes, are taken into account in the numerical calculations. The variation trends in binding energy corresponding to the four impurity states as functions of structural dimension and Al composition differ in nanorods and nanotubes when the impurity moves toward the interface between the Ga₂O₃ and (Al<i>_x</i>Ga_1−<i>x</i>)₂O₃ layers. The quantum confinement due to the structural geometry has a considerable influence on the probability density of the impurity states as well as the impurity binding energy. The numerical results will pave the way toward theoretical simulation of the electron states in rapidly developing β-Ga₂O₃ low-dimensional material systems for optoelectronic device applications.