No, a dog barking cannot cause a ferret to die.
Here is why:
Barking is noise. While loud noises can be stressful for ferrets, a dog's bark alone is unlikely to cause serious harm or death. Ferrets are small and delicate but also surprisingly resilient. Excessive ongoing stress can negatively impact a ferret's health, but a single bark or a few barks are unlikely to cause significant stress.
It is important to consider:
A dog barking in a threatening way toward a ferret could cause fear and stress. Ferrets with underlying health issues may be more susceptible to stress. It is crucial to keep ferrets and dogs separate and supervised to prevent physical interactions that could lead to injury.
If you are concerned about your ferret's health or well-being, consult a veterinarian.