Infinite dimensional spaces consisting of sequences that do not converge to zero

Given a map $f \colon E \longrightarrow F$ between Banach spaces (or Banach lattices), a set $A$ of $E$-valued bounded sequences, ${\bf x} \in A$ and a vector topology $\tau$ on $F$, we investigate the existence of an infinite dimensional Banach space (or Banach lattice) containing a subsequence of ${\bf x}$ and consisting, up to the origin, of sequences $(x_j)_{j=1}^\infty$ belonging to $A$ such that $(f(x_j))_{j=1}^\infty$ does not converge to zero with respect to $\tau$. The applications we provide encompass the improvement of known results, as well as new results, concerning Banach spaces/Banach lattices not satisfying classical properties and linear/nonlinear maps not belonging to well studied classes.