  The Born-Infeld $f(T)$ theory is reduced from the Born-Infeld determinantal
gravity in Weitzenb\"ock spacetime. We investigate a braneworld scenario in
this theory and obtain an analytic domain wall solution by utilizing the
first-order formalism. The model is stable against the linear tensor
perturbation. It is shown that the massless graviton is localized on the brane,
but the continuous massive gravitons are non-localized and will generate a tiny
correction with the behavior of ${1}/{(k r)^{3}}$ to the Newtonian potential.
The four-dimensional teleparallel gravity is recovered as an effective infrared
theory on the brane. As a physical application, we consider the
(quasi-)localization property of spin-1/2 Dirac fermion in this model.
