Can a change in magnetic field change the impedance of a coil
This depends on the inductor. If it is air cored, no.

However many inductors have some ferromagnetic core material, iron or ferrite, which have a much higher permeability, typically giving you hundreds of times as much inductance from the same winding.

(There are other reasons for ferromagnetic cores - they direct the mgnetic flux where you want it, for example into the rotor in a motor, or prevent stray magnetic field affecting other circuits) But the main effect is to  multiply the magnetic field strength.

However any ferromagnetic material will saturate at a specific field strength - about 0.3 Tesla for a typical ferrite or just over 1 Tesla for iron. The onset of saturation can be gradual, or relatively sudden, and the details depend on the ferrite or iron alloy composition. It is the portion of the B-H curve where it flattens out.

The answer to the question is that, at saturation, increased current results in less (or no) increase in magnetic field strength - and this is measurable as a reduction in inductance.

This is exploited in saturable reactors, sometimes known as "magnetic amplifiers" where changing a DC bias current, to control saturation, changes the gain of a circuit.