  Let $\overline{k}$ be a fixed algebraic closure of $k$. When the finite place
$v$ is of degree one, we show that all $\overline{k}$-linear relations among
$v$-adic Carlitz multiple polylogarithms at algebraic points arise from
$k$-linear relations among these values of the same weight. As an application,
we establish a function field analogue of Furusho-Yamashita's conjecture for
$v$-adic multiple zeta values whenever the degree of the place $v$ is one.
