Infinitely presented graphical small cancellation groups are acylindrically hyperbolic

We prove that infinitely presented graphical $C(7)$ and $Gr(7)$ small cancellation groups are acylindrically hyperbolic. In particular, infinitely presented classical $C(7)$-groups and, hence, classical $C'(\frac{1}{6})$-groups are acylindrically hyperbolic. We also prove the analogous statements for the larger class of graphical small cancellation presentations over free products. We construct infinitely presented classical $C'(\frac{1}{6})$-groups that provide new examples of divergence functions of groups.