  In this article, we prove the integrality of $v$-adic multiple zeta values
(MZVs). For any index $\mathfrak{s}\in\mathbb{N}^r$ and finite place $v\in
A:=\mathbb{F}_q[\theta]$, Chang and Mishiba introduced the notion of the
$v$-adic MZVs $\zeta_A(\mathfrak{s})_v$, which is a function field analogue of
Furusho's $p$-adic MZVs. By estimating the $v$-adic valuation of
$\zeta_A(\mathfrak{s})_v$, we show that $\zeta_A(\mathfrak{s})_v$ is a $v$-adic
integer for almost all $v$. This result can be viewed as a function field
analogue of the integrality of $p$-adic MZVs, which was proved by
Akagi-Hirose-Yasuda and Chatzistamatiou.
