Consider $X$ a Banach space and its continuous dual $X^*$. We know that if the dual norm $\|\cdot\|^*$ of $X^*$ is Fr'echet differentiable then $X$ is reflexive (e.g., see Theorem 8.6 in

Fabian, Marián; Habala, Petr; Hájek, Petr; Montesinos Santalucía, Vicente; Pelant, Jan; Zizler, Václav, Functional analysis and infinite-dimensional geometry, New York, NY: Springer. ix, 451 p. (2001). ZBL0981.46001.)

If the norm $\|\cdot\|$ of $X$ is Frechet differentiable, does it imply that $X$ is reflexive? If not, could we construct a non-reflexive space $X$ such that $\|\cdot\|$ is Frechet differentiable?