The notion of a *strongly regular * Banach space was introduced and studied in
[Some topological and geometrical structures in Banach spaces, Ghoussoub et al., Memoirs of the American Mathematical Society, (1987), No. 378], see
here.

A Banach space $X$ is called *strongly regular* if for every $\varepsilon>0$ and every nonempty
convex bounded subset $C\subset X$ there exist positive reals $t_1,\dots, t_k$ with $\sum_{i=1}^n t_i=1$ and
nonempty relatively weak-open subsets $U_1,\dots ,U_n \subset C$ such that the norm diameter of $\sum_{i=1}^n t_i U_i$ is less than $\varepsilon$.

Let $C^k(M)$ be the Banach space of the $k$-times continuously differentable real-valued functions on a smooth compact manifold $M$ with the usual norm. I wish to show that $C^k(M)$ is not strongly regular (because it is an assumption in a theorem I would like to quote). Is this known?

I suspect that no Banach space that contains an isomorphic copy of $c_0$ is strongly regular. Is this true?

Is there a slick way to see that $C^k(M)$ contains a copy of $c_0$? I think I can prove by hand by embedding $c_0$ to $C([0,1])$, and then embedding the latter into $C^k(M)$ by integrating $k$ times and using spherical coordinates, but I would rather quote a reference.

Disclaimer: Banach spaces in not my area of expertise.