The n-time generalization of the Tangherlini solution is considered. The equations of geodesics for the metric are integrated. For n = 2 it is shown that the naked singularity is absent only for two sets of parameters, corresponding to the trivial extensions of the Tangherlini solution. The motion of a relativistic particle in a multi-temporal background is considered. This motion is governed by the gravitational mass tensor. Some generalizations of the solution, including the multi-temporal analogue of the Myers--Perry charged black hole solution, are obtained.