  Inspired by the BTZ formalism, we discuss the Maxwell-$f(T)$ gravity in
(2+1)-dimensions. The main task is to derive exact solutions for a special form
of $f(T)=T+\epsilon T^2$, with $T$ being the torsion scalar of
Weitzenb$\ddot{\mbox{o}}$ck geometry. To this end, a triad field is applied to
the equations of motion of charged $f(T)$ and sets of circularly symmetric
non-charged and charged solutions have been derived. We show that, in the
charged case, the monopole-like and the ${\it ln}$ terms are linked by a
correlative constant despite of known results in teleparallel geometry and its
extensions [39]. Furthermore, it is possible to show that the event horizon is
not identical with the Cauchy horizon due to such a constant. The singularities
and the horizons of these black holes are examined: they are new and have no
analogue in literature due to the fact that their curvature singularities are
soft. We calculate the energy content of these solutions by using the general
vector form of the energy-momentum within the framework of $f(T) $ gravity.
Finally, some thermodynamical quantities, like entropy and Hawking temperature,
are derived.
