Not necessarily. The hole will be stationary relative to Mars so you could just send a little base station with radio and power that drops a paper towel roll sized module with an unspooling data/power link cable and your science stuff into the ice. It can either bore its way through like a TBM (hard but good scientific value since you can measure environment on the way down) or just have it heat itself and let gravity drop it through (no moving parts but less scientific value since you're altering the environment around the probe). When you hit water stop unspooling the cable and just hang it there by the cable.
I'm sure college students in Maine or Montana would be able to pool their ice fishing expertise and come up with some working prototypes if given a sufficient quantity of beer.
It's fun to play armchair engineer.
Maybe we just need to go full Bruce Willis. Somebody get Ben Affleck to sing Leaving on a Jet Plane.
In seriousness, it sounds like a fascinating problem. If anyone has seen the drills they use to make holes for piles for new condo/office towers—they're probably wider than needed in this case (not actually sure of that)— the drill (bit?) is extendable, but the rig itself is enormous. Those are mobile units.
Otherwise something like a semi-permanent or permanent unfolding oil-rig type machine would be needed?
None of this helps with the weight problem. I wonder if there's a more clever option.
It would be similar to preparing a shallow mine.
After the hole is made you get to clean it up to obtain decent samples though.
Whether this would save size or mass is to be seen, a digging drone would preferably be small though very sturdy and dense to survive landslides.
I think a drilling drone was proposed, but communications posed complicated. That is, communicating to a base rover to route to Earth. I could picture some kind of high-tensile wire on a spool connected to the digging drone. It might not be the highest speed communication, but if it at least proved reliable, that could work, no?
1- make a rover of standard weight and size an put it in the desired place, as before. Deploy a light but strong flat platform around.
2-Look around. What do you see? Probably a lot of heavy rocks.
3-Pick some rocks of the appropriate mass, or a bag of sand
4-Load stuff of your choice over rover or put it in the flatform or in a special area designed to be filled until reaching the desired weight. Yes, like a truck.
5- Now that your rover is 8x heavier, start drilling
6- Finished?. Now remove the bags of sand or the rocks one to one.
It starts to become complicated...