The interference of stray electrons (also called ''electron clouds'') with accelerator beams is important in modern intense-beam accelerators, especially those with beams of positive charge. In magnetic wigglers, used, for instance, for transverse emittance damping, the intense synchrotron radiation produced by the beam can generate an electron cloud of relatively high density. In this paper the complicated dynamics of electron clouds in wigglers is examined using the example of a wiggler in the Cornell Electron Storage Ring Test Accelerator experiment at the Cornell Electron Storage Ring. Three-dimensional particle-in-cell simulations with the WARP-POSINST computer code show different density and dynamics for the electron cloud at locations near the maxima of the vertical wiggler field when compared to locations near the minima. Dynamics in these regions, the electron cloud distribution vs longitudinal position, and the beam coherent tune shift caused by the wiggler electron cloud will be discussed.