Let $T$ be a bounded linear operator acting on a complex Banach space. Suppose that $T$ has spectral radius strictly less than $1$. If we introduce an analytic perturbation to $T$, $s\mapsto T_s$ for $|s|<\epsilon$ (with $T_0 = T$), then by upper-semicontinuity of the spectrum, assuming that $\epsilon$ is sufficiently small, the spectral radius of each $T_s$ is less than $\rho$ for some $\rho <1$. My question is the following:

Is it possible to find $K>0$ and $\rho <\rho '<1$ such that $$\|T_s^n\|\le K (\rho ')^n,$$ for all $|s|<\epsilon$ and $n\in \mathbb{Z}_{\ge 0}$?

This certainly seems like it should be true but I can't find a proof - I think I'm missing something obvious. Any help would be greatly appreciated - cheers!