Given an open subset U of a complex Banach space E, a weight v on U and a complex Banach space F, let $$H^\infty _v(U,F)$$
H
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–Bollobás property for $$H^\infty _v(U,F)$$
H
v
∞
(
U
,
F
)
($$WH^\infty $$
W
H
∞
-BPB property, for short). A result of Lindenstrauss type with sufficient conditions for $$H^\infty _v(U,F)$$
H
v
∞
(
U
,
F
)
to have the $$WH^\infty $$
W
H
∞
-BPB property for every space F is stated. This is the case of $$H^\infty _{v_p}(\mathbb {D},F)$$
H
v
p
∞
(
D
,
F
)
with $$p\ge 1$$
p
≥
1
, where $$v_p$$
v
p
is the standard polynomial weight on $$\mathbb {D}$$
D
. The study of the relations of the $$WH^\infty $$
W
H
∞
-BPB property for the complex and vector-valued cases is also addressed as well as the extension of the cited property for mappings $$f\in H^\infty _v(U,F)$$
f
∈
H
v
∞
(
U
,
F
)
such that vf has a relatively compact range in F.