We propose a numerical method for modeling highly deformable nonlinear incompressible solids that conserves the volume locally near each node in a finite element mesh. Our method works with arbitrary constitutive models, is applicable to both passive and active materials (e.g. muscles), and works with simple tetrahedra without the need for multiple quadrature points or stabilization techniques. Although simple linear tetrahedra typically suffer from locking when modeling incompressible materials, our method enforces incompressibility per node (in a one-ring), and we demonstrate that it is free from locking. We correct errors in volume without introducing oscillations by treating position and velocity in separate implicit solves. Finally, we propose a novel method for treating both object contact and self-contact as linear constraints during the incompressible solve, alleviating issues in enforcing multiple possibly conflicting constraints.