The Stark problem in quantum-confined geometries is challenging in high fields. In particular, the localization of electrons near sample boundaries is hard to accurately capture in perturbation methods. We analyze the problem for spherical quantum dots using a combination of numerical diagonalization and analytical perturbation approaches. Closed-form expressions for polarizabilities and hyperpolarizabilities of arbitrary states are obtained. In addition, a hypergeometric resummation ansatz replicating the correct high-field behavior is constructed. We find that a simple resummation approach is superior to even 14th-order perturbation series.