The first thing that came to my mind was a loose cable (similar to BAretired's chain), but then I thought "that's a system, not a material". I agree with BAretired. I've never heard of a material that can deflect without taking load and then all of a sudden become stiff when the load hits a certain point.
How about a sponge? It wouldn't offer 0 resistance, but in tension it would be very soft until the fibres were stretched out. In compression it would behave similarly until it was fully consolidated.