Are magnetic hysteresis losses relevant to alternating currents flowing in a wire?

Say we have an AC in a magnetically lossy material, like iron. Because of iron's relatively high permeability, skin effect will be more pronounced than it is in say, copper, so this iron wire isn't so great a conductor, practically speaking.

However, skin effect arises due to eddy currents, which are themselves due to the time varying magnetic field due to the time varying current. So, this suggests there is a magnetic field at work, and there's the potential for magnetic hysteresis to be an additional source of loss in the material.

Yet, I'm not sure if this is true or not, if somehow the geometry of the conductor and the fields around it make this a non-issue. I would think that if a time-varying magnetic field exists inside the iron, then there would be hysteresis losses. Is this true? Does such a field exist?

To be clear: the issue is with iron as an electric conductor, not as part of a magnetic circuit such as a transformer core as is the more common application.

$$\delta = \sqrt{\frac{2\rho}{\omega \mu}}$$