Suppose \(\Omega\) is a Lipschitz domain in \(\mathbb{R}^n, n \geq 2\), \(g : \partial \Omega \to S ^{n-1}\) is a Lipschitz map of degree \(d\), and for each \(p \in [n-1,n)\), \(u_p : \Omega \to S ^{n-1}\) is a \(p\)-energy minimizing map with \(u _p |_{\partial \Omega} = g\). The authors generalize all main results of the earlier paper [R. Hardt and F. Lin, Calc. Var. Partial Differ. Equ. 3, No. 3, 311-341 (1995; Zbl 0828.58008)] where the planar case \(n = 2\) was considered under assumptions that \(\Omega\) and \(g\) are \({\mathcal C}^2\)-smooth. As a consequence, an analog of the complete boundary regularity known in the \({\mathcal C}^2\)-smooth case is obtained. Namely, the \(p\)-energy minimizing map \(u_p\) has no boundary singularities for Lipschitz \(\Omega\) and \(g\) whenever \(p\) is close enough to \(n\).