  Let $K$ and $k$ be $p$-adic fields. Let $L$ be the composite field of $K$ and
a certain Lubin-Tate extension over $k$ (including the case where
$L=K(\mu_{p^{\infty}})$). In this paper, we show that there exists an
explicitly described constant $C$, depending only on $K,k$ and an integer $g
\ge 1$, which satisfies the following property: If $A_{/K}$ is a
$g$-dimensional CM abelian variety, then the order of the $p$-torsion subgroup
of $A(L)$ is bounded by $C$. We also give a similar bound in the case where
$L=K(\sqrt[p^{\infty}]{K})$. Applying our results, we study bounds of orders of
torsion subgroups of some CM abelian varieties over number fields with values
in full cyclotomic fields.
