  In this paper, we study the thick brane scenario constructed in the recently
proposed $f(T,\mathcal{T})$ theories of gravity, where $T$ is called the
torsion scalar, and $\mathcal{T}$ is the trace of the energy-momentum tensor.
We use the first-order formalism to find analytical solutions for models that
include a scalar field as a source. In particular, we describe two interesting
cases in which, in the first, we obtain a double-kink solution, which generates
a splitting in the brane. In the second case, proper management of a kink
solution obtained generates a splitting in the brane intensified by the torsion
parameter, evinced by the energy density components satisfying the weak and
strong energy conditions. In addition, we investigate the behavior of the
gravitational perturbations in this scenario. The parameters that control the
torsion and the trace of the energy-momentum tensor tend to shift the massive
modes to the core of the brane, keeping a gapless non-localizable and stable
tower of massive modes and producing more localized massless modes.
