A circuit/component which can reduce own resistance with increase of current
What you're looking for is called a "shunt regulator". The simplest shunt regulator is a simple zener diode:







simulate this circuit &ndash; Schematic created using CircuitLab

This circuit functions as a voltage-limited halfwave rectifier. The drawback is that all of the load current (in both directions) flows through the zener diode, and it must be rated for this.

The first refinement we can make to this is to use a full-wave rectifier:







simulate this circuit

But while this doubles the current available to the circuit, it also doubles the dissipation of the zener. But we can use a power transistor to off-load most of that dissipation:







simulate this circuit

If the current gain of the transistor is, say, 50, then only 1/50 of the load current passes through the zener. The rest is handled by the transistor (which will probably need a hefty heatsink).

Now, if you need a truly isolated supply, then we can use the voltage-limited AC waveform at the AC terminals of the bridge rectifier to drive a 1:1 transformer, like this:







simulate this circuit