<p>Peridynamics (PD) has proven to be a promising theory to describe the behavior of materials allowing for singularities and fracture. The classical PD theory restricts the Poisson ratio. To address this issue, Continuum-kinematics-inspired Peridynamics (CPD) has been proposed as a variationally consistent formulation that can capture the Poisson effect exactly. Due to its geometrically exact nature, CPD does not suffer from zero-energy modes and displacement oscillations, making it an ideal nonlocal elasticity framework for large deformations. In a two-dimensional setting, CPD builds upon one-neighbor and two-neighbor interactions. One-neighbor interactions capture length-associated elasticity between pairs of points, equivalent to the original PD formalism. The two-neighbor interactions of CPD recover area-associated elasticity between triplet of points. This contribution provides for the first time a correspondence material model for CPD in a two-dimensional setting such that it recovers a well-established compressible neo-Hookean energy density of nonlinear elasticity at large deformations. At small deformations, the proposed model reduces to classical isotropic linear elasticity. The theory is illustrated via a series of numerical examples.</p>