We present analytic results for a special dimer model on the {em non-bipartite} and {em non-planar} checkerboard lattice that does not allow for parallel dimers surrounding diagonal links. We {em exactly} calculate the number of closed packed dimer coverings on finite checkerboard lattices under periodic boundary conditions, and determine all dimer-dimer correlations. The latter are found to vanish beyond a certain distance. We find that this solvable model, despite being non-planar, is in close kinship with well-known paradigm-setting planar counterparts that allow exact mappings to $mathbb{Z}_2$ lattice gauge theory.
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