Prime-to-p torsion lifts uniquely at good reduction ($E(K)[q^{\infty }]\cong \tilde{E}(k)[q^{\infty }],q\ne p$)

For good reduction of an elliptic curve over a complete local field of residue characteristic $p$, smoothness and the Hensel lemma make reduction surjective. Its kernel is a formal group on the maximal ideal, where multiplication by $q^m$, $q\ne p$, is bijective by prime-to-residue-characteristic multiplication on a formal group. Thus there is no nonzero $q$-power torsion in the kernel. Given a lift $P$ of a reduced point killed by $q^m$, choose the unique kernel point $R$ with $q^{m}R=q^{m}P$. Then $P-R$ is a torsion lift. This proves both surjectivity and uniqueness of torsion lifting. For finite residue field these groups are finite.