I'm planning a short course on few topics and applications of nonlinear functional analysis, and I'd like a reference for a quick and possibly self-contained construction of a structure of a Banach differentiable manifold for the space of continuous mappings $C^0(K,M)$, where $K$ is a compact topological space (even metric if it helps) and $M$ is a (finite dimensional) differentiable manifold.
A construction of a differentiable structure of Banach manifold for this space can be found e.g. in Lang's book Fundamentals of differential geometry (1999). The main tools are the exponential map and tubular nbds (having fixed a Riemannian structure on $M$. This is OK but I believe there should be something even more basic.
Does anybody have a reference for alternative constructions (not necessarily elementary) ?