I read an article where it is said: $E_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$
where $E$ is an elliptic curve over $\mathbb{Q}_p$ and $E_1(\mathbb{Q}_p)=\{P\in E(\mathbb{Q}_p):\tilde{P}=\tilde{O}\}$.
The author says that the proof is in "Arithmetic of elliptic curves" by J.Silverman, at page 191, but here it is said:
If $E$ is an elliptic curve over $\mathbb{Q}_p$ and $\hat{E}$ is the formal group, then:
$$E_1(\mathbb{Q}_p)\approx \hat{E}(p\mathbb{Z}_p)$$
So I do not know a good reference for the proof of $E_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$.
At the beginning of the formal group law chapter Silverman shows how to define it from the Weierstrass equation and gives some hints why the obtained series $w\in \Bbb{Z}[[z]]$ will be useful when letting $z\in p\Bbb{Z}_p $ (that he calls $\mathcal{M}$) and that as set $E_{formal}(p \Bbb{Z}_p) = \{ (z,w(z)), z\in p\Bbb{Z}_p\}$
In the local field - reduction $\bmod \pi$ - chapter he shows that $E_1(\Bbb{Q}_p) \cong E_{formal}(p \Bbb{Z}_p)$ where $\cong$ is how we constructed the formal group and $E_1(\Bbb{Q}_p)= \{ (x,y) \in \Bbb{Q}_p^2,\not \in \Bbb{Z}_p^2, y^2=x^3+ax+b\} \cup O$ the points whose projective coordinate will be in $[p\Bbb{Z}_p:1:p\Bbb{Z}_p]$ whose reduction will be $[0:1:0]\in E(\Bbb{F}_p)$
The group isomorphism $E_{formal}(p \Bbb{Z}_p) \to p \Bbb{Z}_p$ is the formal logarithm, at the end of the formal group law chapter (for finite extensions $K/\Bbb{Q}_p$ the isomorphism is $E_{formal}(\pi_K^r O_K) \to \pi_K^r O_K$ where $r > \frac1{p-1}v(p)/v(\pi_K)$)