Let \( F \) be a closed subspace of \( \ell _p \) (with \( p \in [1, \infty ] \) ) containing \( c_0 \) . In this paper, we prove that every finite-dimensional subspace contained in \( F {\setminus } Z(F) \) can be extended to a closed subspace of maximal dimension still contained in \( F {\setminus } Z(F) \) . This result strengthens the work of Cariello and Seoane-Sepúlveda, published in (J Funct Anal 266:3797–3814, 2014), by providing new insights into the linear structural enrichment within the complement of the set of sequences with at most finitely many zeros. We also show that such extendability fails for certain infinite-dimensional subspaces, thereby revealing intrinsic limitations to the closed linear structure of \( F {\setminus } Z(F) \) . Among other results, we prove that the set \( Z(X) \) —where \( X \in \{c_0, \ell _p\} \) , with \( p \in [1, \infty ] \) —is [ \(\mathcal {S}\) ]-lineable for every subspace \( S \subset \ell _\infty \) . We also establish, for the first time, its maximal pointwise lineability in this setting. Finally, we show that all of the above results remain valid even in the case \( p \in (0,1) \) , significantly broadening the scope of the previous contributions.
