The blocks with five irreducible characters

Let $G$ be a finite group, $p$ a prime and $B$ a Brauer $p$-block of $G$ with defect group $D$. We prove that if the number of irreducible ordinary characters in $B$ is $5$ then $D\cong C_5,C_7,D_8$ or $Q_8$, assuming that the Alperin–McKay conjecture holds for $B$.