= Solution
Apply the <spectral-radius maximum principle> to the entire <Banach-space-valued holomorphic function> $f(z)=a+zb$ on the <closed> disk $|z|\le\varepsilon$. On its <boundary> the assumed estimate gives $r(f(z))\le C\varepsilon$. Therefore
$$
0\le r(a)=r(f(0))\le C\varepsilon
$$
for every $\varepsilon>0$. Letting $\varepsilon\downarrow0$ proves
$$
\boxed{r(a)=0.}
$$
The puncture at $z=0$ in the hypothesis causes no difficulty: the bound is used only on the nonzero <boundary> circle.
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