Suppose that $(A, ∥ · ∥_A)$ and $(B, ∥ · ∥_B)$ are two Banach algebras such that $B$ is an ideal in $A$ and $∥·∥_A ≤ ∥·∥_B$. We know that each Banach $A$-bimodule $X$ is also a Banach $B$-bimodule. But the converse is not true in general!
Do you have an example of a $B$-bimodule $X$ that is not an $A$-bimodule, where the $B$-module action on $X$ is the $A$-module action on $X$ restricted to $B$?