  Given an open subset $U$ of a complex Banach space $E$, a weight $v$ on $U$
and a complex Banach space $F$, let $H^\infty_v(U,F)$ denote the Banach space
of all weighted holomorphic mappings from $U$ into $F$, endowed with the
weighted supremum norm. We introduce and study a version of the
Bishop--Phelps--Bollob\'as property for $H^\infty_v(U,F)$ ($WH^\infty$-BPB
property, for short). A result of Lindenstrauss type with sufficient conditions
for $H^\infty_v(U,F)$ to have the $WH^\infty$-BPB property for every space $F$
is stated. This is the case of $H^\infty_{v_p}(\mathbb{D},F)$ with $p\geq 1$,
where $v_p$ is the standard polynomial weight on $\mathbb{D}$. The study of the
relations of the $WH^\infty$-BPB property for the complex and vector-valued
cases is also addressed as well as the extension of the cited property for
mappings $f\in H^\infty_v(U,F)$ such that $vf$ has relatively compact range in
$F$.
