The authors study complete immersed surfaces contained in a slab of a warped product \(\mathbb{R}\times_\rho {\mathbb{P}}^2\), where \(\mathbb{P}^2\) is complete with nonnegative Gaussian curvature. Under certain restrictions on the mean curvature of the surface, the authors prove that such an immersion does not exist or must be a leaf of the trivial totally umbilical foliation \(t\in {\mathbb{R}}\to \{t\}\times {\mathbb{P}}^2\).