The answer to the question is almost surely negative (as almost always in Banach space theory) but I cannot find a relevant example.
Is there an example of an infinite-dimensional Banach space $X$ such that $X^{**}$ does not contain infinite-dimensional reflexive subspaces?
Note that such a space must be HI-saturated.
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.
