The Nike factory shop is described as selling products at a lower mark-up to attract customers, even though it purchases them at the same price as traditional retailers. This business model, where a manufacturer sells its own products directly to the public at reduced prices, is characteristic of a factory outlet. Factory outlets are a specific type of off-price retailer. Off-price retailers sell merchandise at lower prices than conventional retailers, often by buying goods at less than regular wholesale prices or by operating on lower margins, as described in the question (lower mark-up). The correct option is c). The final answer is $\boxed{\text{c. off-price retailer}}$