  We compute the Kaluza-Klein spectrum of the non-supersymmetric
SO(3)$\,\times\,$SO(3)-invariant AdS$_4$ vacuum of 11-dimensional supergravity,
whose lowest-lying Kaluza-Klein modes belong to a consistent truncation to
4-dimensional ${\cal N}=8$ supergravity and are stable. We show that,
nonetheless, the higher Kaluza-Klein modes become tachyonic so that this
non-supersymmetric AdS$_4$ vacuum is perturbatively unstable within
11-dimensional supergravity. This represents the first example of unstable
higher Kaluza-Klein modes and provides further evidence for the AdS swampland
conjecture, which states that there are no stable non-supersymmetric AdS vacua
within string theory. We also find 27 infinitesimal moduli amongst the
Kaluza-Klein modes, which hints at the existence of a family of
non-supersymmetric AdS$_4$ vacua.
