We study the existence of infinite-dimensional vector spaces in the sets of NORM-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations PP of the set {1,…,n}, there exists a closed infinite-dimensional Banach subspace of the space of nn-linear forms on ℓ1ℓ1 such that, for all nonzero elements BB of such a subspace, the Arens extension associated to the permutation σσ of BB is NORM-attaining if and only if σσ is an element of PP. We also study the structure of the set of NORM-attaining nn-linear forms on c0c0.