  Let $\mathbb F$ be an algebraically closed field of characteristic $p\ge 0$,
which is complete with respect to a non-Archimedean absolute value. Let $V$ be
a projective subvariety of $\mathbb P^M(\mathbb F)$. In this paper, we will
prove some second main theorems for non-Archimedean meromorphic maps of
$\mathbb F^m$ into $V$ intersecting a family of hypersurfaces in $N-$subgeneral
position with truncated counting functions.
