An isogeometric membrane element based on the non-uniform rational basis spline (NURBS) model is presented that accounts for membrane wrinkling based on tension-field theory. First, the element is validated by means of a benchmark problem involving a partly wrinkled membrane. It is then applied to the large deformation of a thin membrane structure using a two-stage procedure that combines dynamic relaxation and Newton-Raphson iteration. A simple technique is introduced that takes advantage of the geometrical symmetry of an isogeometric analysis model by using a "one-sided" open knot vector to treat the continuity of the membrane surface with respect to the symmetry plane. Because NURBS has many suitable features for representing complex geometries, it enriches the function space of the membrane element. Consequently, the characteristic mechanical responses of membranes, such as deep folding, are captured appropriately by the present isogeometric membrane element. In addition, a numerical example demonstrates that the convergence rate of the isogeometric membrane analysis with respect to refinement of the discretization is much better when tension-field theory is introduced in the analysis.