**Question:** Let $\mathbb{F}$ be an algebraically closed field of characteristic zero and let $\mathrm{Ch}_{\mathbb{F}}$ be the category whose objects are chain complexes (of $\mathbb{F}$-modules) and whose morphisms are chain maps. Is it true that every object in $\mathrm{Ch}_{\mathbb{F}}$ is injective? If so, does it remain true if $\mathbb{F}$ is not algebraically closed? What if $\mathbb{F}$ has positive characteristic?

On one hand this seems to good to be true, but on the other hand things are often exceedingly nice when dealing with vector spaces.

**Edit:** removed a flawed proof. Thanks darij for spotting the error.