  We solve two main questions on linear structures of (non-)norm-attaining
Lipschitz functions. First, we show that for every infinite metric space $M$,
the set consisting of Lipschitz functions on $M$ which do not strongly attain
their norm and the zero contains an isometric copy of $\ell_\infty$, and
moreover, those functions can be chosen not to attain their norm as functionals
on the Lipschitz-free space over $M$. Second, we prove that for every infinite
metric space $M$, neither the set of strongly norm-attaining Lipschitz
functions on $M$ nor the union of its complement with zero is ever a linear
space. Furthermore, we observe that the set consisting of Lipschitz functions
which cannot be approximated by strongly norm-attaining ones and the zero
element contains $\ell_\infty$ isometrically in all the known cases. Some
natural observations and spaceability results are also investigated for
Lipschitz functions that attain their norm in one way but do not in another,
for several norm-attainment notions considered in the literature.
