Yes, there are such spaces. To see this, first note that Joram Lindenstrauss showed that for every separable Banach space $Y$ there exists a Banach space $X$ such that $X^{\ast\ast}$ is separable and $X^{\ast\ast}/X$ is isomorphic to $Y$. Apply this result with $Y$ a separable Banach space that is not isomorphic to a subspace of a separable dual space (e.g., $c_0$ or $L_1$; see for example Theorem 6.3.7 from Albiac and Kalton's book *Topics in Banach Space Theory* for details). Then, for such $Y$, the space $X$ prescribed by Lindenstrauss' construction cannot be isomorphic to a dual space. Indeed, if it were, then $X^{\ast\ast}/X$ would be isomorphic to a subspace of $X^{\ast\ast}$ by virtue of the fact that that every dual space is complemented in its bidual (in particular, $X^{\ast\ast}$ would be isomorphic to $(X^{\ast\ast}/X)\oplus X$), and thus $Y$ would also be isomorphic to a (complemented) subspace of the separable dual space $X^{\ast\ast}$ - a contradiction.

(As it happens, the aforementioned book of Albiac and Kalton includes Lindenstrauss' result in the chapter with the title 'Important Examples of Banach Spaces'.)