Hello, and welcome to BackYard Chickens. Sorry to hear you're duck is feeling unwell.
Her symptoms are fairly non-specific to any disease, in other words, there are too many things that can cause what you're seeing, so it's hard for me to even suggest what might be the problem. If you could answer some of these questions, that will help.
How old is she? Where did you acquire her? What are you feeding her? Is she laying? Has she had any problems laying in the past, or has she laid any "oddly-shaped" eggs lately? Besides the lethargy part, and not eating, are there any other symptoms we should know about (?) Coughing, respiratory problems, walking funny, etc? Could you post a few pictures of the duck in question and a few pictures of her poop?
There is obviously something wrong going on here, so the bird needs to be set in a warm, soft padded, calm, quiet area, with feed and water provided. If the coop is getting cold, I would suggest bringing her inside. Birds that are sick tend not to eat as much, which lowers their metabolic energy levels, which subsequently leads to a lowered body temperature (hypothermia). I would start weighing her every morning, so if she is losing weight, we can track exactly how much she is, so we know if tube feeding needs to be set in place. A 5 to 10% decrease in weight grants supplemental tube feeding.
Contrary to what I said earlier in this post, her symptoms do slightly resemble a bird suffering from laying problems. That is, if she is actually laying eggs, of course. In my experience, the number one cause of sudden lethargy in grown waterfowl is reproductive problems such as minor egg binding. It tends to resolve itself, although a nice warm soak in an epsom salt bath may help, as well as an administration of a calcium supplement such as Calcium Citrate, tums, or calcium Gluconate 23%.
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