  In this paper, the concept of selective real manifolds is extended. It is
proved that the product of two selective Banach manifolds is a selective Banach
manifold. The notion of the $\alpha$--level differentiation of the mappings
between selective Banach manifolds is presented. Basic properties of $(r,
\alpha)$--differentiable maps are studied. Tangent space for at a point of a
selective Banach manifold is considered.
