  Given a pointed metric space $M$, we study when there exist $n$-dimensional
linear subspaces of $\operatorname{Lip}_0(M)$ consisting of strongly
norm-attaining Lipschitz functionals, for $n\in\mathbb{N}$. We show that this
is always the case for infinite metric spaces, providing a definitive answer to
the question. We also study the possible sizes of such infinite-dimensional
closed linear subspaces $Y$, as well as the inverse question, that is, the
possible sizes of the metric space $M$ given that such a subspace $Y$ exists.
We also show that if the metric space $M$ is $\sigma$-precompact, then the
aforementioned subspaces $Y$ need to be always separable and isomorphically
polyhedral, and we show that for spaces containing $[0,1]$ isometrically, they
can be infinite-dimensional.
