  We study Lagrangian immersions in the nearly K\"ahler $\mathbb{S}^6$ which
are warped product manifolds of a $1$-dimensional base and a surface. Apart
from the totally geodesic ones, they are either of constant sectional curvature
$\frac{1}{16}$ or they satisfy equality in Chen's inequality, in which case the
immersion is given explicitly.
