Is it possible to define a structure of differentiable (smooth) manifold on the Hilbert cube $[0,1]^\mathbb{N}$ ?

Has it been done in the literature?

In textbooks, only the Banach case is treated, but the Hilbert cube has countable dimension, and a vector space with countable dimension is not complete (although the Hilbert cube is complete, because of compacity), this is a problem for the tangent space. However, can something be done? maybe with some restrictions?