Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective? The same question for $\ell_\infty$-module $c_0$. Both these modules are not dual, so standard arguments via flatness doesn't work.

Injectivity is understood in the following sense: A Banach module $J$ over Banach algebra $A$ is called injective if for any admissible bounded morphism of $A$-modules $\xi:X\to Y$ and any bounded morphism of $A$-modules $\varphi:X\to J$ there exists a bounded morphism of $A$-modules $\psi:Y\to J$ such that $\psi\xi=\varphi$. Note, a bounded morphism of $A$-modules is called admissible if it admits a left inverse bounded linear operator.