In this paper for a given Banach, possibly infinite dimensional, manifold M we focus on its iterated tangent bundle TrM, r ∈ N ∪ {∞}. First we endow TrM with a canonical atlas using that of M. Then the concepts of vertical and complete lifts for fuctions and vector fields on TrM are defined which they will play an pivotal rule in our next studies i.e. complete lift of sprays. Afterward we supply T∞M with a generalized Fr´echet manifold structure and we will show that any vector field (spray or connection) on TiM, i ∈ N, can be lifted to a vector field (spray or connection respectively) on T∞M. Finally, despite of the natural difficulties with non-Banach modelled manifolds, we will discuss about the ordinary differential equations on T∞M including integral curves and geodesics.
