Do objects sink in ice?

If the article you are referring to is this one, then the planes didn't sink into the ice but were buried by blizzards.

Ice does exhibit ductile flow at stress of around 1 to 10MPa, but this pressure is equivalent to around 100 to 1000 tons per square metre and this is far above the stresses normally produced by objects resting on ice. Glaciers flow because the stress is concentrated at the interface between the ice and the rock, and this can produce the enormous stresses required.

So the answer to your question is that no, objects don't sink into ice under their own weight (unless your object is made from neutronium). However freeze thaw cycles can give this impression.

I say yes, things can sink in ice. Here's why: there's a thin layer of liquid on the surface of ice that is why ice is slippery. When you put a body on the surface of ice, it'll keep displacing the layer of liquid, getting deeper and deeper, and will eventually sink completely,