We study the space of maps from the classifying space of a finite p-group to theBorel construction of a finite group of Lie type G in characteristic p acting on itsbuilding. The first main result is a description of the homology with Fp-coefficients,showing that the mapping space, up to p-completion, is a disjoint union indexedover the group homomorphism up to conjugation of classifying spaces of centralizersof p-subgroups in the underlying group G. We complement this description bydetermining the actual homotopy groups of the mapping space. These resultstranslate to descriptions of the space of maps between a finite p-group and theuncompleted classifying space of the p-local finite group coming from a finite groupof Lie type in characteristic p, providing some of the first results in this uncompletedsetting.