  We study 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 Lipschitz norm (i.e.\ the supremum defining the Lipschitz norm is
a maximum). Extending previous results, we prove that this set is not norm
dense when $M$ is a length space (or local) or when $M$ is a closed subset of
$\mathbb{R}$ with positive Lebesgue measure, providing new examples which have
very different topological properties than the previously known ones. On the
other hand, we study the linear properties which are sufficient to get
Lindenstrauss property A for the Lipschitz-free space $\mathcal{F}(M)$ over
$M$, and show that all of them actually provide the norm density of
$\operatorname{SNA}(M,Y)$ in the space of all Lipschitz maps from $M$ to any
Banach space $Y$. Next, we prove that $\operatorname{SNA}(M,\mathbb{R})$ is
weakly sequentially dense in the space of all Lipschitz functions for all
metric spaces $M$. Finally, we show that the norm of the bidual space of
$\mathcal{F}(M)$ is octahedral provided the metric space $M$ is discrete but
not uniformly discrete or $M'$ is infinite.
