Tate uniformization of nonsplit semistable elliptic curves

A form of this is contained in Silverman, second book, Chapter V, Corollary 5.4. I guess that the image of Gm' in E (at the level of Q_p-points) may have index 2.

Most of it makes sense. Elliptic curves with non-split reduction can be analytically uniformized by the norm torus. There is a "nice" picture of this using the Berkovich spectrum for a non-split torus. I have my doubts about the statement concerning rational points - you should have a Galois cohomology exact sequence.