  In this paper, we introduce and study a Lipschitz version of the
Bishop-Phelps-Bollob\'as property (Lip-BPB property). This property deals with
the possibility of making a uniformly simultaneous approximation of a Lipschitz
map $F$ and a pair of points at which $F$ almost attains its norm by a
Lipschitz map $G$ and a pair of points such that $G$ strongly attains its norm
at the new pair of points. We first show that if $M$ is a finite pointed metric
space and $Y$ is a finite-dimensional Banach space, then the pair $(M,Y)$ has
the Lip-BPB property, and that both finiteness assumptions are needed. Next, we
show that if $M$ is a uniformly Gromov concave pointed metric space (i.e.\ the
molecules of $M$ form a set of uniformly strongly exposed points), then $(M,Y)$
has the Lip-BPB property for every Banach space $Y$. We further prove that this
is the case for finite concave metric spaces, ultrametric spaces, and H\"older
metric spaces. The extension of the Lip-BPB property from $(M,\mathbb R)$ to
some Banach spaces $Y$ and some results for compact Lipschitz maps are also
discussed.
