Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E$ an elliptic curve over $K$ with good ordinary reduction.

The p-adic Tate module $T_p(E)$ is (after tensoring with $\mathbb{Q}_p$) a 2-dimensional $\mathbb{Q}_p$-representation of $\mathop{\mathrm{Gal}}(\bar{K}/K)$.

It is reducible: the kernel of reduction to the residue field is an invariant line.

Does $T_p(E) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$ contain another invariant line?