Let $E$ be an elliptic curve defined over a local field $K_{\mathfrak{p}}$. It's obvious that we can choose a model such that $E$ is defined over the ring of integer $\mathcal{O}_{K_{\frak{p}}}$. Thus, we may define the reduction map $\pi_{\frak{p}}$ from $E(K_{\frak{p}})$ to $E(k)$, where $k$ is the residue field w.r.t $\frak{p}$. Note that $E(k)$ may be a singular curve.
Is $\pi_{\frak{p}}$ always surjective? If not, can we get some concrete examples?