No date is given
The Bible never supplies a line such as, “2 Samuel was written in this year.” So the honest answer is that we cannot name an exact date. The book is mainly concerned with what happened in David's reign: his rise as king, his rule from Jerusalem, his sin with Bathsheba, family grief, rebellion, and the closing years of his life.
Scripture does give a clear framework for the period it describes. David began to reign at thirty and ruled for forty years, first in Hebron and then in Jerusalem (2 Samuel 5:4-5). His death is recorded later in 1 Kings 2:10. Those events belong to David's lifetime, but they do not by themselves tell us when the final book was compiled.
David was thirty years old when he began to reign, and he reigned forty years. In Hebron he reigned over Judah seven years and six months: and in Jerusalem he reigned thirty and three years over all Israel and Judah.
Samuel did not write all of it
The name “Samuel” can make this question confusing. Samuel played a major part in David's early story, but he died before David became king over all Israel (1 Samuel 25:1). That means he could not have written the later events found in 2 Samuel, including David's final years.
The division between 1 and 2 Samuel was not part of the original Hebrew book. Together, they tell one continuing story, beginning with Samuel's birth and ending near the close of David's reign. The second half carries Samuel's name because it belongs to that larger story, not because he personally wrote every chapter.
Earlier records stood behind the story
A helpful clue comes in the account of David's death. It says that records of David's acts were written by Samuel the seer, Nathan the prophet, and Gad the seer (1 Chronicles 29:29). Nathan and Gad were both active during David's reign, so they could record events Samuel never saw.
This verse does not tell us precisely who arranged 2 Samuel into its final form or when that happened. Still, it shows that written prophetic records about David existed. Many readers therefore understand 2 Samuel as a book shaped from trustworthy records preserved from David's time and brought together later.
Now the acts of David the king, first and last, behold, they are written in the book of Samuel the seer, and in the book of Nathan the prophet, and in the book of Gad the seer,
Why the uncertainty matters less than the message
It is reasonable to ask when a biblical book was written. Yet 2 Samuel does not ask us to place our confidence in a modern publication date. It shows God keeping his promise to establish David's kingdom, while also taking David's sin seriously. God made a covenant with David concerning his house and kingdom (2 Samuel 7:12-16).
Read the book with both questions in view: what happened in David's life, and what does it reveal about God? Its account is strikingly honest. David is honoured as king, but his failures bring real sorrow. That honesty is one reason the book continues to speak so powerfully.
And when thy days be fulfilled, and thou shalt sleep with thy fathers, I will set up thy seed after thee, which shall proceed out of thy bowels, and I will establish his kingdom. He shall build an house for my name, and I will stablish the throne of his kingdom for ever. I will be his father, and he shall be my son. If he commit iniquity, I will chasten him with the rod of men, and with the stripes of the children of men: But my mercy shall not depart away from him, as I took it from Saul, whom I put away before thee. And thine house and thy kingdom shall be established for ever before thee: thy throne shall be established for ever.