  The generalized Schwarzschild spacetimes are introduced as warped manifolds
where the base is an open subset of $\mathbb{R}^2$ equipped with a Lorentzian
metric and the fiber is a Riemannian manifold. This family includes physically
relevant spacetimes closely related to models of black holes. The generalized
Schwarzschild spacetimes are endowed with involutive distributions which
provide foliations by lightlike hypersurfaces. In this paper, we study
spacelike submanifolds immersed in the generalized Schwarzschild spacetimes,
mainly, under the assumption that such submanifolds lie in a leaf of the above
foliations. In this scenario, we provide an explicit formula for the mean
curvature vector field and establish relationships between the extrinsic and
intrinsic geometry of the submanifolds. We have derived several
characterizations of the slices, and we delve into the specific case where the
warping function is the radial coordinate in detail. This subfamily includes
the Schwarzschild and Reissner-Nordstr\"om spacetimes.
