So water will transition from liquid to solid and back through a phase change which consumes or produces energy. So any melting will cool the drink assuming it's warmer than 0 degrees C.
So let's take as our starting point a mass of ice uniformly 0 degrees C. Regardless of geometry, any cooling of whisky will inevitably produce a phase change to liquid water in the exact amount required by the energy absorbed by the ice. So you get 0 degree ice, 0 degree water, and cooler whiskey. Assuming a 0 degree ice ball geometry is irrelevant.
Of course water can be cooler than 0! So let's assume some really cool ice and do some thought experiment geometries. Take super thin sheets of ice for example. The total amount of energy available to cool the drink without melting is the specific heat of the ice times the temperature difference from zero (there might be some non-linearity here but I doubt it's relevant) times the volume of the ice. So if you have enough volume and cold enough ice you should be able to achieve a 0 degree drink at which point you have no more cooling and therefore no melting and no water.
Unless the heat is transferred out of the water unevenly, say a large mass of it with a central core inaccessible from the outside liquid. Then it would matter how fast heat travels from the interface to the core relative to how fast heat travels from the drink to the interface. This probably has something to do with the outside liquid being turbulent and the inside solid being a lattice but I'm not sure.
So, pure speculation here, I'd guess a large ball would have smaller surface to volume and so would cool more slowly for equivalent volume, but you get a large volume so you might be okay assuming the cold core of the large ice ball does the job of cooling the surface faster than the drink can heat it up.
It seems to be it would be more effective to get very small ice chunks, a ton of them, make them really cold, then you have lots of surface area and if you do the math you could approximately guarantee no melting and you'd cool fast. Then strain out the chunks and put whatever you want in there since it's already 0 (maybe something cool looking you can condescend about?)
But whatever you do don't put it in a glass you grab with your whole hand, the air is a really good insulator, glass much less so and palms work like a circulating hot-fluid heat exchanger.