Cohomology class of a non torsion point

Let $k$ be a finitely generated fields of positive characteristic $p>0$.
Let $E$ be an ordinary elliptic curve over $k$ with a non torsion, non zero, rational point $x$.

By the kummer sequence we get a map:

where $T_p(E)$ is $\varprojlim_n E(k^{sep})[p^n]$.

Is it possible to say that $\phi(x)$ is non zero and non torsion? If not, is it possible find some condition on $E$ or $k$ to ensure this?

If think it is always true that the image of $x$ is non torsion in $\varprojlim_n H^1_{flat}(k,E[p^n])$.