The answer is that there is indeed an example of such space. This is established in Theorem 6.27 of:

Argyros, Spiros A.; Arvanitakis, Alexander D.; Tolias, Andreas G. Saturated extensions, the attractors method and hereditarily James tree spaces. Methods in Banach space theory, 1–90, London Math. Soc. Lecture Note Ser., 337, Cambridge Univ. Press, Cambridge, 2006.

This book chapter is freely available online at https://arxiv.org/pdf/0807.2392.pdf

Briefly, Theorem 6.27 establishes the existence of an infinite dimensional Banach space $\mathfrak{X}_{\mathcal{F}_s'}$ whose dual does not contain any infinite dimensional reflexive subspace. Since $\mathfrak{X}_{\mathcal{F}_s'}$ admits a boundedly complete basis, denoted $(e_n)$, $\mathfrak{X}_{\mathcal{F}_s'}$ is isomorphic to a dual space. In particular, the closed linear span in $\mathfrak{X}_{\mathcal{F}_s'}^\ast$ of the coordinate functionals $(e_n^\ast)$ biorthogonal to $(e_n)$ provides an example of such a space as requested in OP's question.