Thank you everyone for your hints, especially
@WannaBeHillBilly 's coverage of various possibilities... This is what has happened since I described the problem:
- We visited the vet on day 4 (not possible over the weekend, not the avian specialist anyway), she took X-rays, saw nothing special, agreed that there are no palpable lumps on the body from the head all the way to the vent... gave us general analgesic and a nourishing powder which is used dissolved in water in an emergency situation
- We started feeding this solution and the analgesic by syringe twice daily
- I dunked her head several times every day to keep the nares etc in good condition
- We noticed that she does drink on her own, reluctantly but it does happen; she seems somewhat interested in eating pellets and grains from the bowl but always pulls back in the middle of the neck movement and doesn't eat after all
- The duck stopped laying but that only happened on day 5 since noticing the issue so I believe / hope that this is not an egg related thing; I consider the stopped production to be the consequence of being undernourished
- We still had most of the antibiotic bottle from Junior's recovery so we started using that today, day 6; we also upgraded from just the emergency powder to a mash of cooked peas, the powder, some olive oil and apple vinegar in order to "unplug" the duck's bowels in case that's what this was all about
... And it worked! After feeding the combined mash and water, after a while we finally saw a single little poo, greenish-whitish and soft but not slimy or liquid kind of soft. It ooked just fine.
Afterwards I let her go to her flock (from the dog crate where we do the syringe thing and keep her inside for a while). She drank a lot from the various basins and kiddie pools. She moved her neck more freely - not yet fully OK but in a much improved range of motion. She checked her feathers and preened a bit which is great.
When she's in a good enough condition that I can no longer catch her we'll consider the problem resolved
All in all this was finally a good day.
We'll continue with the pea mash, the analgesic, the antibiotic, and also add some dry yeast to the mash.
I have a theory which I don't consider super likely but it's a possibility -- that this situation is maybe due to pawpaw (asimina triloba) leaves. We have several pawpaw trees in the orchard; the twigs and the leaves are very powerful biochemically, they can even be used as an insecticide. Whereas I did not notice ducks going for them while they were still green, now that they have dropped it could be that pieces of the leaves got eaten while the ducks rummaged around for worms etc. In time the effect could accumulate. I'll rake them up.
