We consider the problem of a small body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible steady Navier–Stokes equations in an exterior domain in a half-space, with appropriate boundary conditions on the wall, the body and at infinity. In this paper, we first prove in a very general setup the existence of weak solutions for the problem with the body. Then, we show that any such solution can be truncated and then extended to provide a weak solution for a simplified problem where the body is replaced by a (small) source term with compact support. This simplified problem was already shown to possess strong solutions. We then prove a weak–strong uniqueness theorem to show the uniqueness of solutions for the simplified problem. Finally, we show that this also implies the uniqueness of solutions for the problem of the moving body which proves that the solutions of both problems have the same asymptotic behavior at infinity.