Summary: We study approximation properties of \(hp\)-finite element subspaces of \({\mathbf H}(\text{div},\Omega)\) and \({\mathbf H}(\text{rot},\Omega)\) on a polygonal domain \(\Omega\) using Brezzi-Douglas-Fortin-Marini (BDFM) or Raviart-Thomas (RT) elements. Approximation theoretic results are derived for the \(hp\)-version finite element method on non-quasi-uniform meshes of quadrilateral elements with hanging nodes for functions belonging to weighted Sobolev spaces \({\mathbf H}_{\omega}^{s,\ell}(\Omega)\) and the countably normed spaces \({\mathcal B}_{w}^{\ell}(\Omega)\). These results culminate in a proof of the characteristic exponential convergence property of the \(hp\)-version finite element method on suitably designed meshes under similar conditions needed for the analysis of the \({\mathbf H}^{1}(\Omega)\) case. By way of illustration, exponential convergence rates are deduced for mixed \(hp\)-approximation of flow in porous media.