We investigate the solutions of black holes in $f(T)$ gravity with nonlinear power-law Maxwell field, where $T$ is the torsion scalar in teleparalelism. In particular, we introduce the Langranian with diverse dimensions in which the quadratic polynomial form of $f(T)$ couples with the nonlinear power-law Maxwell field. We explore the leverage of the nonlinear electrodynamics on the space-time behavior. It is found that these new black hole solutions tend towards those in general relativity without any limit. Furthermore, it is demonstrated that the singularity of the curvature invariant and the torsion scalar is softer than the quadratic form of the charged field equations in $f(T)$ gravity and much milder than that in the classical general relativity because of the nonlinearity of the Maxwell field. In addition, from the analyses of physical and thermodynamic quantities of the mass, charge and the Hawking temperature of black holes, it is shown that the power-law parameter affects the asymptotic behavior of the radial coordinate of the charged terms, and that a higher-order nonlinear power-law Maxwell field imparts the black holes with the local stability.
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