We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken H(curl) space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to a posteriori error analysis of Maxwell's equations discretized with Nédélec's finite elements of arbitrary order. The resulting estimators are locally efficient, polynomial-degree-robust, and inexpensive since the quasi-equilibration only goes over edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The estimators become guaranteed when the regularity pickup constant is explicitly known. Numerical experiments illustrate the theoretical findings.