Rational torsion points of abelian varieties over a large extension of a local field

We extend the following theorem of H. Imai in several ways:
If  A is an abelian variety with potentially good reduction
over a finite extension  K  of  Q_p, then it has only
finitely many rational torsion points over the maximal
p-cyclotomic extension of  K. In particular, we prove
the finiteness over  K(K^{1/p^\infty}). It has applications
in Iwasawa theory.