The paper illustrates the application of the position-based finite element formulation to the nonlinear analysis of hyperelastic membranes. The formulation adopts the nodal position vectors in the current configuration as the main unknowns of the problem instead of the nodal displacements. As a result, simple analytical expressions are deduced for the secant and tangent stiffness matrices, which enable straightforward implementation of any hyperelastic constitutive law. Here, Ogden’s material model is adopted in conjunction with Pipkin’s concept of relaxed energy. In this way, the effects of wrinkling and buckling are automatically taken into account and treated as material nonlinearities. As an illustrative example, a polyurethanic rectangular membrane in bending is analyzed. The obtained results are in good agreement with previous results of the literature.