I read an article where it is said: $E_1(\mathbb{Q}_p)\cong \mathbb{Z}_p$ 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 there it is said:

If $E$ is an elliptic curve over $\mathbb{Q}_p$ and $\hat{E}$ is the formal group, then:

$$E_1(\mathbb{Q}_p)\cong \hat{E}(p\mathbb{Z}_p)$$

So I don't know a good reference for the proof of $E_1(\mathbb{Q}_p)\cong \mathbb{Z}_p$.