2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116347Various definitions of C^k-maps on open subsets of finite-dimensional vector spaces over a complete valued field have been proposed in the literature. We show that the C^k-maps considered by Schikhof and De Smedt coincide with those of Bertram, Glockner and Neeb. By contrast, Ludkovsky's C^k-maps need not be C^k in the former sense, at least in positive characteristic. We also compare various types of Holder differentiable maps on finite-dimensional and metrizable spaces.extended preprint version (46 pages) with additional appendices B and CFunctional AnalysisGeneral Mathematics26E30 (Primary) 26E20; 46A16; 46G05; 46S10 (Secondary)text