Dyneema rope will do well. It's a low-stretch marine rope used on sailboats and other applications. Although it's low-stretch, it still stretches more than wire rope. The numbers for Young's modulus are about 4 or 4.5 million psi for the Dyneema and about 11 or 12 million psi for the steel cable, so if you double the nominal diameter for the Dyneema, you'll be okay for low-stretch.
And Dyneema's strength is so great that it simply won't be an issue. It's very light. I think it'll float.
The caveats and gotchas are:
1. The Dyneema is slippery and knots aren't reliable. Worse, they'll break at relatively low loads. Only a proper splice should be used, such as this one, which is fun to make:
http://www.neropes.com/SPL_12Strand_EyeSpliceBrummel.aspx
Personally, I can't splice this if it's thinner than 1/8" thick, and that's difficult. Thicker is easier.
Incidentally, the buried end needs to be well-tapered and buried at least 44 times the diameter. The easy way to taper it is to bring it out of the cover (bunch up the cover) below where you want the end to be, cut the taper on the individual strands, and then pull the cover back over them.
2. The Dyneema is sensitive to stress concentrations and won't develop its strength around sharp edges or small radius objects. Ideally you need something like 8 times the rope diameter for the diameter of the object the Dyneema is wrapping around. Smaller gives proportionally lower strength, and since the rope strength in the oversize diameter is high, you might be able to get away with it in some applications.
3. Dyneema creeps. If there's a steady load it'll gradually stretch. You wouldn't use it for control cables, for example, since they generally need tension.
4. It's pretty abrasion-resistant and fairly UV resistant. In both cases it'll wear or soften on the exposed fibers and you can see and feel that. But it's not as durable as wire rope in that regard.
5. Being a low-stretch rope, it doesn't absorb much energy when loaded. You would not want to use it for climbing, for example, if you fell, the shock would be very severe.
You can get Dyneem from places like
http://www.apsltd.com/c-1492-amsteel-blue-samson.aspx, under the trade name "Amsteel Blue."
It's good stuff, when used with knowledge of its characteristics.
Dave