In 

Stegall, C. Banach spaces whose duals contain $\ell_1(\Gamma)$ with applications to the study of dual $L_1(\mu)$ spaces. Trans. Amer. Math. Soc. 176 (1973), 463–477 

Stegall proved that $\ell_2$ is isometrically isomorphic to a norm one complemented subspace of $X^*$ when $X= (\sum_{n=1}^\infty \ell_2^n)_1$, yet $\ell_2$ does not embed into $X$.

It would be interesting to have an example of this phenomena with $X^*$ separable.