New applications of extremely regular function spaces

Let $L$ be an infinite locally compact Hausdorff topological space. We show that extremely regular subspaces of $C_0(L)$ have very strong diameter $2$ properties and, for every real number $\epsilon$ with $0<\epsilon <1$, contain an $\epsilon$-isometric copy of $c_0$. If $L$ does not contain isolated points they even have the Daugavet property, and thus contain an asymptotically isometric copy of ${\mathrm{\ell }}_1$.