In this paper we explore the five-dimensional f(T,B) teleparallel modified gravity with f1(T,B)=T+k1Bn1 and f2(T,B)=B+k2Tn2 in the brane scenario. Asymptotically, the bulk geometry converges to an AdS5 spacetime whose cosmological constant is produced by parameters that control torsion and the boundary term. The analysis of the energy density condition reveals a splitting brane process satisfying the weak and strong energy conditions for some values of the parameters n1,2 and k1,2. In addition, we investigate the behavior of the gravitational perturbations in this scenario. In the bulk, the torsion keeps a gapless nonlocalizable and stable tower of massive modes. Inside the brane core, the torsion produces new barriers and potential wells leading to small amplitude massive modes and a massless mode localized for some values of the parameters n1,2 and k1,2.