To add to Parvin, this usage of "duck" is still common in the north of England, among some older people, and would be less common among young people. We have "bab" in Birmingham, used for both men and women in a similar way and is common even among some younger people, perhaps from their 30s. They're terms of endearment. Older people can often use terms of endearment more naturally than young people so they may seem more old-fashioned than they really are.

Some (older) BrE speakers use "duck" or "duckie" (or "ducks") as a familiar term for the addressee, in conversation. The speaker is almost always female. Some (again, older) male BrE speakers use "old duck" as a term of affection for their wives. "Old duck" can also be used as a synonym for "old chap"; but this is very rare. These usages belong to very particular contexts and kinds of speaker; also, they now have a slightly old-fashioned air.