  We investigate $f(T)$ theory coupled with a nonlinear source of
electrodynamics, for a spherically symmetric and static spacetime in $4D$. We
re-obtain the Born-Infeld and Reissner-Nordstrom-AdS solutions. We generalize
the no-go theorem for any content that obeys the relationship
$\mathcal{T}^{\;\;0}_{0}=\mathcal{T}^{\;\;1}_{1}$ for the energy-momentum
tensor and a given set of tetrads. Our results show new classes of solutions
where the metrics are related through $b(r)=-Na(r)$. We do the introductory
analysis showing that solutions are that of asymptotically flat black holes,
with a singularity at the origin of the radial coordinate, covered by a single
event horizon. We also reconstruct the action for this class of solutions and
obtain the functional form $f(T) = f_0\left(-T\right)^{(N+3)/[2(N+1)]}$ and
$\mathcal{L}_{NED} = \mathcal{L}_0\left(-F\right)^{(N+3)/[2(N+1)]}$. Using the
Lagrangian density of Born-Infeld, we obtain a new class of charged black holes
where the action reads $f(T) = -16\beta_{BI} \left[1 - \sqrt{1 +
(T/4\beta_{BI})}\right]$.
