This article considers a plane strain gradient plasticity theory of the Gurtin--Anand model [M. Gurtin, L. Anand, \emph{A theory of strain gradient plasticity for isotropic, plastically irrotational materials Part I: Small deformations}, J. Mech. Phys. Solids extbf{53} (2005), 1624--1649] for an isotropic material undergoing small deformation in the absence of plastic spin. It is assumed that the system of microstresses is purely dissipative, so that the free energy reduces to a function of the elastic strain, while the microstresses are only related to the plastic strain rate and gradient of the plastic strain rate via the constitutive relations. The plane strain problem of the Gurtin--Anand model for a purely dissipative process gives rise to elastic incompressibility. A weak formulation of the flow rule is derived, making the plane strain problem suitable for finite element implementation.