  Holm introduced $m$-free $\ell$-arrangements which is a generalization of
free arrangements, while he asked whether all $\ell$-arrangements are $m$-free
for $m$ large enough. Recently Abe and the author verified that this question
is in the negative when $\ell\geq 4$. In this paper we verify that
$3$-arrangements $\mathscr{A}$ are $m$-free and compute the $m$-exponents for
all $m\geq |\mathscr{A}|+2$, where $|\mathscr{A}|$ is the cardinality of
$\mathscr{A}$. Hence Holm's question is in the positive when $\ell=3$. Finally
we prove that $3$-dimensional Weyl arrangements of types A and B are $m$-free
for all $m\geq 0$.
