For an arbitrary infinite-dimensional Banach space 𝔛, we construct examples of strongly-measurable 𝔛-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly differentiable; thus, for these functions the Lebesgue Differentiation Theorem fails rather spectacularly. We also relate the degree of nondifferentia-bility of the indefinite Pettis integral to the cotype of 𝔛, from which it follows that our examples are reasonably sharp.