Do you know a reference where i can read the proof of this theorem?:
Let $p$ be an odd prime and let $E$ be an elliptic curve over $\mathbb{Q}_p$ for which its reduction $\tilde{E}$ is an elliptic curve too. Furthermore, assume that there is a homomorphism $s : \tilde{E}(\mathbb{F}_p) \longrightarrow E(\mathbb{Q}_p)$ such that $\psi\circ s = id_{|{\tilde{E}(\mathbb{F}_p)}}$, where $\psi$ is the reduction map.
Then we have $E(\mathbb{Q}_p)\approx \mathbb{Z}_p \times \tilde{E}(\mathbb{F}_p)$.