Measurements of K(892)0⁎ and ϕ(1020) resonance production in Pb–Pb and pp collisions at s_NN = 5.02 TeV with the ALICE detector at the Large Hadron Collider are reported. The resonances are measured at midrapidity (|y| < 0.5) via their hadronic decay channels and the transverse momentum (p_T) distributions are obtained for various collision centrality classes up to p_T = 20 GeV/c. The p_T-integrated yield ratio K(892)0⁎/K in Pb–Pb collisions shows significant suppression relative to pp collisions and decreases towards more central collisions. In contrast, the ϕ(1020)/K ratio does not show any suppression. Furthermore, the measured K(892)0⁎/K ratio in central Pb–Pb collisions is significantly suppressed with respect to the expectations based on a thermal model calculation, while the ϕ(1020)/K ratio agrees with the model prediction. These measurements are an experimental demonstration of rescattering of K(892)0⁎ decay products in the hadronic phase of the collisions. The K(892)0⁎/K yield ratios in Pb–Pb and pp collisions are used to estimate the time duration between chemical and kinetic freeze-out, which is found to be ∼ 4–7 fm/c for central collisions. The p_T-differential ratios of K(892)0⁎/K, ϕ(1020)/K, K(892)0⁎/π, ϕ(1020)/π, p/K(892)0⁎ and p/ϕ(1020) are also presented for Pb–Pb and pp collisions at s_NN = 5.02 TeV. These ratios show that the rescattering effect is predominantly a low-p_T phenomenon.