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There is no absolute agreement about what the Book of David (Zaboor or Zabur) is referring to. The majority opinion is that it refers to the Book of Psalms, which is a part of the Bible. The majority of the Psalms are traditionally written by David.
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Every Ethiopian Orthodox priest starts with the study of the book of David, my father being one of them. It's available in Ethiopia but it's in Ge'ez, an almost extinct language now used solely by the Ethiopian Orthodox church.
I have not heard of any translations but I know a tiny bit of the language and it is strictly structured for the Bible: Every word and character is built around the Bible so when it gets translated things might get lost in translation and as every one knows is very very dangerous.
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David was not the sole author of any books of the Bible, but:
He authored most (but not all) of the Psalms, a Jewish Hymnbook that is the Largest book of the Bible.
He is the primary character in the second half of 1 Samuel and all of 2 Samuel, plus its duplicate 1 Chronicles.
According to this source, the Books of David are often assumed to be the Psalms, but are not so universally held. Likewise, the Books of Isa are often assumed to be the New Testament, but could refer to an unknown Gospel authored by Jesus himself. In any event, these are not canonical books.
A better understanding of the Islamic views on these books would of course be found on Islam.SE, but you'd know that :)