  Let $k$ be a perfect field of characteristic $p > 2$. We extend the
equivalence of categories between Fontaine-Laffaille modules and $\mathbb{Z}_p$
lattices inside crystalline representations with Hodge-Tate weights at most
$p-2$ of Fontaine and Laffaille to the situation where the base ring is the
power series ring over the Witt vectors $ W(k)[\![ t_1, \cdots , t_d]\!]$ and
where the base ring is a $p$-adically complete ring that is \'etale over the
Tate Algebra $W(k)\langle t_1^{\pm 1}, \cdots , t_d^{\pm 1}\rangle$.
