Let $K$ be a local field with uniformizer $\pi$ and residue field $k$ and let be $E$ an elliptic curve defined over $K$. We have the exact sequence of abelian groups
$$0 \rightarrow E_1(K) \rightarrow E_0(K) \rightarrow \widetilde{E}_{\text{ns}}(k) \rightarrow 0$$
where $E_1(K)$ is the kernel of reduction mod $\pi$, $E_0(K)$ contains the points of nonsingular reduction, and $\widetilde{E}_\text{ns}(k)$ is the group of nonsingular $k$-rational points on the reduced curve $\widetilde{E}$.
I'm aware of some examples where this sequence splits, but can anything be said in general?