Sublinear growth of 1-cocycles and uniform convexity

Let G be a finitely generated group, let $\pi \text{\hspace{0.17em}⁣}:G\to \mathrm{G}\mathrm{L}(E)$ be a uniformly bounded $c_0$-representation on a superreflexive Banach space $E$, and let $b\text{\hspace{0.17em}⁣}:G\to E$ be a $1$-cocycle for $\pi$. Then $b$ has sublinear growth with respect to the word length. As a corollary we obtain the corresponding Hilbert space statement for strongly mixing unitary representations.