This is impossible. I will prove that if an injective module $M$ contains a decomposable module, then $M$ is decomposable.
By hypothesis, $M$ has a submodule of the form $N_1 \oplus N_2$, where each $N_i$ is nonzero. Because $M$ is injective, it contains a submodule isomorphic to the injective hull $E(N_1 \oplus N_2) = E(N_1) \oplus E(N_2)$ (for this equality, see Lam's Lectures on Modules and Rings, equation (3.39)). Because this is an injective submodule of $M$, we have $M = E(N_1) \oplus E(N_2) \oplus P$ for some submodule $P$. Because both of the injective hulls $E(N_i)$ are nonzero, it follows that $M$ is decomposable.
The positive way to state this result is that an injective module is indecomposable if and only if it is uniform. You can find this statement in the same book by Lam, Theorem 3.52.