The article is devoted to the study of groups of diffeomorphisms and wraps of manifolds over ultra-metric fields of zero and positive characteristic. Different types of topologies are considered on groups of wraps and diffeomorphisms relative to which they are generalized Lie groups or topological groups. Among such topologies, pairwise incomparable ones are also found. Topological perfectness of the diffeomorphism group relative to certain topologies is studied. Theorems on projective limit decompositions of these groups and their compactifications for compact manifolds are proved. Moreover, the existence of one-parameter local subgroups of diffeomorphism groups is proved.