Trapped atomic ions have been used successfully to demonstrate basic elements of universal quantum information processing. Nevertheless, scaling up such methods to achieve large-scale, universal quantum information processing (or more specialized quantum simulations) remains challenging. The use of easily controllable and stable microwave sources, rather than complex laser systems, could remove obstacles to scalability. However, the microwave approach has drawbacks: it involves the use of magnetic-field-sensitive states, which shorten coherence times considerably, and requires large, stable magnetic field gradients. Here we show how to overcome both problems by using stationary atomic quantum states as qubits that are induced by microwave fields (that is, by dressing magnetic-field-sensitive states with microwave fields). This permits fast quantum logic, even in the presence of a small (effective) Lamb-Dicke parameter (and, therefore, moderate magnetic field gradients). We experimentally demonstrate the basic building blocks of this scheme, showing that the dressed states are long lived and that coherence times are increased by more than two orders of magnitude relative to those of bare magnetic-field-sensitive states. This improves the prospects of microwave-driven ion trap quantum information processing, and offers a route to extending coherence times in all systems that suffer from magnetic noise, such as neutral atoms, nitrogen-vacancy centres, quantum dots or circuit quantum electrodynamic systems.