Summary: A multi-dimensional field model with (at most) one Einstein space of non-zero curvature and \(n\) Ricci-flat internal spaces is considered. The action contains arbitrary numbers of dilatonic scalar fields \(\varphi^a\) and antisymmetric forms \(F_s\) of both electric and magnetic types, as they appear in the weak field limit of theories like M-theory (associated with \(p\)-branes). The problem setting covers various models with field dependence on a single space-time coordinate, in particular, homogeneous cosmologies, static, spherically symmetric and Euclidean models. Exact solutions are obtained when the \(p\)-brane dimensions and the dilatonic couplings obey orthogonality conditions in the minisuperspace. For spherically symmetric solutions, conditions for black hole and wormhole existence are formulated. As in four dimensions, wormholes can only exist with negative energy density (such as pure imaginary scalar fields). For black holes, an analogue of no-hair theorems for \(F\)-forms is obtained; it is shown that even in spaces with multiple times a black hole may only exist with its unique, one-dimensional time; an infinite value of the Hawking temperature is explicitly shown to imply a curvature singularity at an assumed horizon, and such cases among extreme black hole solutions are indicated.