Modified Kadomtsev-Petviashvili equation describing nonlinear dynamics of nearly one-dimensional wave structures in dusty plasma above illuminated part of the Moon in the situation in which localization along magnetic-field vector is much stronger than along other directions is derived. The equation differs from the ordinary Kadomtsev-Petviashvili equation by the nonlinear term being non-analytical. Modified Kadomtsev-Petviashvili differs from generalizations of the Kadomtsev-Petviashvili equation in which nonlinearity retains the same form as in the ordinary Kadomtsev-Petviashvili equation but higher-order corrections for dispersion are taken into account. An analytical expression governing one-dimensional soliton solution to the modified Kadomtsev-Petviashvili equation is obtained. The solution differs from the well-known one-dimensional soliton solutions to the Korteweg-De Vries and ordinary Kadomtsev-Petviashvili equations. Stability analysis of the one-dimensional soliton solution showed that it is stable. Possible applications of the discussed solitons from the point of view of description of the so-called transient lunar phenomena representing short-lived light, changes in color or appearance on the surface of the Moon are discussed.