An integer parallelotope with small surface area

We prove that for any $n\in \mathbb{N}$ there is a convex body $K\subseteq {\mathbb{R}}^n$ whose surface area is at most $n^{\frac{1}{2}+o(1)}$, yet the translates of $K$ by the integer lattice ${\mathbb{Z}}^n$ tile ${\mathbb{R}}^n$.