  We study the density of the set $\operatorname{SNA}(M,Y)$ of those Lipschitz
maps from a (complete pointed) metric space $M$ to a Banach space $Y$ which
strongly attain their norm (i.e.\ the supremum defining the Lipschitz norm is
actually a maximum). We present new and somehow counterintuitive examples, and
we give some applications. First, we show that $\operatorname{SNA}(\mathbb
T,Y)$ is not dense in ${\mathrm{Lip}}_0(\mathbb T,Y)$ for any Banach space $Y$,
where $\mathbb T$ denotes the unit circle in the Euclidean plane. This provides
the first example of a Gromov concave metric space (i.e.\ every molecule is a
strongly exposed point of the unit ball of the Lipschitz-free space) for which
the density does not hold. Next, we construct metric spaces $M$ satisfying that
$\operatorname{SNA}(M,Y)$ is dense in ${\mathrm{Lip}}_0(M,Y)$ regardless $Y$
but which contains an isometric copy of $[0,1]$ and so the Lipschitz-free space
$\mathcal F(M)$ fails the Radon--Nikod\'{y}m property, answering in the
negative a posed question. Furthermore, an example $M$ can be produced failing
all the previously known sufficient conditions to get the density of strongly
norm attaining Lipschitz maps. Finally, among other applications, we prove that
given a compact metric $M$ which does not contains any isometric copy of
$[0,1]$ and a Banach space $Y$, if $\operatorname{SNA}(M,Y)$ is dense, then
$\operatorname{SNA}(M,Y)$ actually contains an open dense subset and
$B_{\mathcal F(M)}=\overline{{\mathrm{co}}}(\operatorname{str-exp}(B_{\mathcal
F(M)}))$. Further, we show that if $M$ is a boundedly compact metric space for
which $\operatorname{SNA}(M,\mathbb R)$ is dense in ${\mathrm{Lip}}_0(M,\mathbb
R)$, then the unit ball of the Lipschitz-free space on $M$ is the closed convex
hull of its strongly exposed points.
