Let $V\_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V\_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum with $V\_0$ isn't closed?

A positive solution would let me answer [this question](https://mathoverflow.net/questions/431740/under-what-conditions-does-a-continuous-linear-map-map-a-closed-subspace-to-a-cl).