Relation between inductor core size and maximum current

[CityBird]

Must a core for a 100mH/10A inductor have a larger cross section than a 100mH/1A inductor?

 

I am confused here. Ampere's law says that the Field strength is proportional to the current flowing though the windings (H= n.I/lm). Core saturation B/H (permeability, μ in fact) at the other hand does not depend on the actual core area. One may logically conclude that a 10A inductor could be assembled with the exact same core, at least if one would be able to fit the thicker wire. This seems counter-intuitive to me.