  Let $X$ be a complex Banach space. We prove that if $L$ is an extremally
disconnected compact Hausdorff topological space, then the pair $(X, C(L))$
satisfies the Bishop-Phelps-Bollob\'as property (BPBp for short). As a
byproduct, we obtain the BPBp for the pair $(X, L^\infty(\nu))$ for any measure
$\nu$. In particular, this settles an unresolved question regarding the BPBp
for the pair $(L^\infty(\mu), L^\infty(\nu) )$ for any two measures $\mu$ and
$\nu$. Finally, we show that $(X,H^\infty(\Omega)$ has the BPBp when $\Omega$
is a multi-connected planar domain bounded by finitely many disjoint analytic
simple closed curves.
