Is the magnetic force an example of a nonconservative force?

Answer 1

Yes. For any vector field, force included, to be conservative, it must be irrotational, meaning that its curl must be zero everywhere. Fortunately (for me at least, since it makes this answer a whole heck of a lot easier to explain), the magnetic force field vector is already commonly expressed as a curl via Maxwell's equations:

∇ X B = μ0J + μ0ε0∂E/∂t,

where B is the magnetic field, μ0 is the permeability of free space, ε0 is the permittivity of free space, J is the current density, E is the electric field, t is time, and ∇ X B is the curl of the magnetic field. Bolded quantities are vectors.

Now, if you don't know a thing about vector calculus, hopefully you at least remembered what I wrote above. For a vector field to be conservative, its curl must be zero everywhere. As you can hopefully see from that above equation, the curl of the magnetic field, ∇ X B, is not zero everywhere. More specifically, if there is a current density, J, within the magnetic field, or if there is a time changing electric field, ∂E/∂t, within the magnetic field, the curl of the magnetic field is not zero, therefore the magnetic force field is nonconservative.

If you get lucky enough to find a situation where there is no current density or time dependent electric field, which is in reality impossible due to electrons providing there own time dependent electric fields, then the magnetic force field would be conservative.