In this paper we consider the Hu-Washizu principle and propose a new dual-mixed finite element method for nonlinear incompressible plane elasticity with mixed boundary conditions. The approach extends a related previous work on the Dirichlet problem and imposes the Neumann (essential) boundary condition in a weak sense by means of an additional Lagrange multiplier. The resulting variational formulation becomes a twofold saddle point operator equation which, for convenience of the subsequent analysis, is shown to be equivalent to a nonlinear threefold saddle point problem. In this way, a slight generalization of the classical Babuška–Brezzi theory is applied to show the well-posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimates. In particular, the classical PEERS space is suitably enriched to define the associated Galerkin scheme. Next, we develop a local problems-based a posteriori error analysis and derive an implicit reliable and quasi-efficient estimate, and a fully explicit reliable one. Finally, several numerical results illustrating the good performance of the explicit a posteriori estimate for the adaptive computation of the discrete solutions are provided.