We present a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a spatial branch-and-bound algorithm. It introduces a new operation, called lifting, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. We discuss conditions under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient, and then present a computational technique based on ellipsoidal calculus that satisfies these conditions. We also analyze the convergence properties of the branch-and-lift algorithm. Finally, we illustrate the practical applicability of branch-and-lift with numerical examples.