  In this paper, we introduce and study a first-order mean-field game obstacle
problem. We examine the case of local dependence on the measure under
assumptions that include both the logarithmic case and power-like
nonlinearities. Since the obstacle operator is not differentiable, the
equations for first-order mean field game problems have to be discussed
carefully. Hence, we begin by considering a penalized problem. We prove this
problem admits a unique solution satisfying uniform bounds. These bounds serve
to pass to the limit in the penalized problem and to characterize the limiting
equations. Finally, we prove uniqueness of solutions.
