  This paper gives some characterizations of the boundedness of the maximal or
nonlinear commutator of the $p$-adic fractional maximal operator $
\mathcal{M}_{\alpha}^{p}$ with the symbols belong to the $p$-adic BMO spaces on
(variable) Lebesgue spaces and Morrey spaces over $p$-adic field, by which some
new characterizations of BMO functions are obtained in the $p$-adic field
context. Meanwhile, Some equivalent relations between the $p$-adic BMO norm and
the $p$-adic (variable) Lebesgue or Morrey norm are given.
