Immersed boundary methods (IBMs) have proven to be valuable tools for the numerical investigation of complex flow configurations and fluid–structure interaction (FSI). Several IBM flavours have been introduced to address different flow settings. Among them, Eulerian methods enforce the no-slip condition on non-conforming walls through flow reconstruction near the body, but are not adequate for moving walls and FSI due to the presence of emerging fluid nodes and difficulties in treating zero-thickness bodies. On the other hand, Lagrangian methods are well suited for moving bodies thanks to the diffusion of the interface, but are known to suffer from large slip errors. Furthermore, when the IBM is implemented within a fractional-step method, the no-slip condition is modified during the projection step. To avoid this, a boundary condition on the walls should be enforced on the elliptic equation, preventing the use of fast Fourier transform (FFT) methods in favour of less efficient solvers. In our work, we propose a novel sharp-interface IBM suitable for moving bodies and zero-thickness walls. In particular, the immersed boundary forcing step is performed before the evaluation of the provisional velocity, and the no-slip condition is enforced on the closest fluid points to the surface through an interpolation of the flow between the bulk and the (possibly) moving wall. The resulting method can treat sharp interfaces (also with zero thickness), satisfying the no-slip condition on the walls up to the precision of the chosen interpolator. To reduce the projection error, the forcing term of the solenoidal projection step is modified consistently with the conservation of mass to take into account the presence of solid walls. However, the discrete Laplace operator is left unchanged, allowing the use of fast Poisson solvers based on FFT algorithms. The novel method is tested against classical benchmarks, such as the flow around a fixed sphere, the FSI of a flapping flag, and highly pressurized bubbles, and the performance is compared with existing IBMs.