Closed linear spaces consisting of strongly norm attaining Lipschitz functionals

Given a pointed metric space M, we study when there exist n-dimensional linear subspaces of Lip(0) (M) consisting of strongly norm-attaining Lipschitz functionals, for n is an element of 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 for a metric space M in order to such a subspace Y exist. 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.