  We show that the Kerr-Schild ansatz can be extended from the metric to the
tetrad, and then to teleparallel gravity where curvature vanishes but torsion
does not. We derive the equations of motion for the Kerr-Schild null vector,
and describe the solution for a rotating black hole in this framework. It is
shown that the solution depends on the chosen tetrad in a non-trivial way if
the spin connection is fixed to be the one of the flat background spacetime. We
show furthermore that any Kerr-Schild solution with a flat background is also a
solution of $f(\mathcal{T})$ gravity.
