One of the central problems in the topology of hyperplane arrangements is determining whether the complement is a $K(π,1)$-space. In this paper, we study Manin--Schechtman arrangements, introduced as higher-dimensional analogs of the braid arrangement, and prove that their complements have non-vanishing higher homotopy groups. Consequently, these arrangements fail to be $K(π,1)$ in a broad range of cases.