The aim of this paper is to study a quasistatic unilateral contact problem between an elastic body and a foundation. The constitutive law is nonlinear and the contact is modelled with a normal compliance condition associated to a unilateral constraint and Coulomb's friction law. The adhesion between contact surfaces is taken into account and is modelled with a surface variable, the bonding field, whose evolution is described by a first-order differential equation. We establish a variational formulation of the mechanical problem and prove an existence and uniqueness result in the case where the friction coefficient is small enough. The technique of proof is based on time-dependent variational inequalities, differential equations and the Banach fixed-point theorem. We also study a penalized and regularized problem which admits at least one solution and prove its convergence to the solution of the model when the penalization and regularization parameter tends to zero.