A new drift-kinetic theory of the ion response to magnetic islands in tokamak plasmas is presented. Small islands are considered, with widths w much smaller than the plasma radius r, but comparable to the trapped ion orbit width ρ_bi. An expansion in w/r reduces the system dimensions from five down to four. In the absence of an electrostatic potential, the ions follow stream lines that map out a ‘drift-island’ structure that is identical to the magnetic island, but shifted by an amount ∼ few ρ_bi. The result qualitatively holds even with the inclusion of the electrostatic potential required for quasi-neutrality. The ion distribution function is flattened across these drift islands, not the magnetic island. For large islands, w ≫ ρ_bi, the effect of the shift is negligible. For small islands, w ∼ ρ_bi, the shift results in a pressure gradient being maintained across the magnetic island. This suppresses the bootstrap current drive for small islands, which is an important new effect, influencing the threshold for neoclassical tearing mode instabilities — a key result for the performance of future tokamaks, including ITER.