In this work we describe a finite element formulation for the approximation of solid mechanics problems using a damage model and considering finite strains. The balance equations are written in a total Lagrangian framework, using the deviatoric component of the second Piola-Kirchhoff stress tensor, the displacement and the pressure as variables. The interest of introducing the pressure as a variable is the possibility to deal with incompressible materials, whereas the convenience of introducing the stress is the improvement in the stress approximation, which may crucial when nonlinear material laws depending on the stress (or the strain) are used. In particular, in this work we use the damage model proposed in [1], which generalises previous isotropic damage models introduced for infinitesimal strains to finite ones. This damage model is combined which a hyperelastic model for the reversible component of the deformation. The three-field formulation was introduced and fully analysed in [2] for the Stokes problem. On the one hand, the interest of interpolating the stress as an independent variable was highlighted in a series of papers started with [3], and later successfully used in numerous works involving both linear and nonlinear constitutive behaviour under the small strain assumption. On the other hand, the authors have extended the three-field formulation to geometrically nonlinear problems in several articles; these formulations are summarised in the review paper [4]. The purpose of this work is precisely to combine these formulations, dealing with problems involving both nonlinear constitutive laws and geometrical nonlinearity.