  In this paper, we provide an infinite metric space $M$ such that the set
$\mbox{SNA}(M)$ of strongly norm-attaining Lipschitz functions does not contain
a subspace which is isometric to $c_0$. This answers a question posed by
Antonio Avil\'es, Gonzalo Mart\'inez Cervantes, Abraham Rueda Zoca, and Pedro
Tradacete. On the other hand, we prove that $\mbox{SNA}(M)$ contains an
isometric copy of $c_0$ whenever $M$ is a metric space which is not uniformly
discrete. In particular, the latter holds true for infinite compact metric
spaces while it does not for proper metric spaces. Some positive results in the
non-separable setting are also given.
