On finite-dimensional vector spaces, we all have a reasonable idea of which functions are likely to be $C^1$ or smooth. When it comes to differentiation on Banach spaces, I find that my `intuition' absolutely fails me and is often completely wrong. Does anyone have an intuitive picture of what it means for a function between Banach spaces to be smooth (or even $C^1$), even just in simple cases like $L^p(\mathbb{R}^n)$ or $W^{1,p}(\mathbb{R}^n)$?

Most of the references I've looked at (Lang, Palais) seem to take a very formal perspective. Does anyone know of a book/paper that emphasizes a more geometric picture?

Motivation: I'm trying to understand Banach manifolds of mappings and smooth maps between them for the construction of moduli spaces in differential geometry. I thought it would be best first to understand what a smooth map between Banach manifolds actually `looks like'.