The article at hand is a generalization of [J. Ludwig, Forum Math. Sigma 5, Paper No. e17, 41 p. (2017; Zbl 1434.11110)] following the same overall strategy. The main theorem proves that the quotient of the Lubin-Tate space at infinite level by the \((n-1,1)\)-block parabolic in \(\mathrm{GL}_n(K)\) is a perfectoid space for any \(n\) and any finite extension \(K\) over \(\mathbb{Q}_p\), while [loc. cit.] dealt with the case of \(n=2\), \(K=\mathbb{Q}_p\). It relies on two main ingredients: a) on the \(\mu\)-ordinary Hasse invariant in the case of unitary Shimura varieties à la Harris-Taylor (as sketched by T. Itō circa 2005); and b) the strategy of P. Scholze’s paper [Ann. Math. (2) 182, No. 3, 945–1066 (2015; Zbl 1345.14031)] to prove that the overconvergent anticanonical tower of certain Shimura varieties is perfectoid. Detailed presentations tailored to this precise case are given for both. An application is given to Scholze’s candidate for mod \(p\) local Langlands (resp. Jacquet-Langlands) correspondence in the guise of a vanishing theorem, as in [J. Ludwig, Forum Math. Sigma 5, Paper No. e17, 41 p. (2017; Zbl 1434.11110)], generalized with a different proof. The appendix by D. Hansen gives a purely local, explicit proof in the context of diamonds for the simpler case \(n=2\), and it shows in particular that the perfectoidness statement for quotient of the Lubin-Tate tower by the \((n-1,1)\)-block parabolic, does not extend to other maximal parabolic of block \((n-d,d)\) for \(d > 1\).