Is it possible for a Banach space to contain strictly its (continuous) dual

Question

Is there an infinite-dimensional Banach space $X$ such that $X^{\star}\subsetneq X$ - i.e. $X^{\star}$ is proper subspace of $X$, both inheriting its linear and topological structure. For example $(c_0)^{\star}=l_1\subsetneq c_0$ however this inclusion is only linear, since these spaces have different topological structure (norms). I am not sure whether the problem is well-posed. Perhaps it turns out that if $X^{\star}\subseteq X$ then $X$ is Hilbert, so a strict inclusion is impossible.