There are a bunch of great books on Banach space geometry but sadly they often do not care very much about Sobolev spaces. There is Example 2.47 in Schuster at al: Regularization Methods in Banach Spaces if you want something directly quotable, although the authors also only point to other works.

However, one can also combine a bunch of results from, say, Cioranescu: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. For example like that:

- For a reflexive Banach space $X$, $X$ is strictly convex [smooth]
*if and only if* $X^*$ is smooth [strictly convex]. This is Corollary II.1.4. So it is enough to establish that $W^{1,p}$ is smooth.
- Theorem I.3.5 says that $X$ is smooth if and only if the norm is G-differentiable on $X \setminus \{0\}$.
- In Chapter II.4 the classical result of Clarkson is established, that the $L^p$ spaces are uniformly convex, in particular strictly convex, so they are also smooth (go back and forth between $L^p$ and its dual). In particular, their norm is G-differentiable except for in $0$.
- The norm on $W^{1,p}$ is a composition of $L^p$ norms and continuous linear operators $W^{1,p} \to L^p$ and thus inherits $G$-differentiability from the $L^p$ ones, so $W^{1,p}$ is smooth and its dual is strictly convex.