Joshua wrote part of the record
The clearest statement comes near the book's end. At Shechem, after Israel renewed its covenant with God, Joshua wrote the words connected with that event in the book of God's law (Joshua 24:26). This gives strong reason to see Joshua as a source for at least part of the book.
The book then tells of Joshua's death, when he was 110 years old (Joshua 24:29). He would not normally have written that account himself. So somebody else, whose name Scripture does not give, likely preserved or completed the final part of the record.
And Joshua wrote these words in the book of the law of God, and took a great stone, and set it up there under an oak, that was by the sanctuary of the Lord.
Details rooted near Joshua's time
Several details sound like they come from people close to the events. After Jericho fell, Rahab and her family were living among Israel when the book's account was recorded (Joshua 6:25). That does not settle the date of every chapter, but it connects the book with the generation that entered Canaan.
Joshua 24:31 also says Israel served the Lord during Joshua's lifetime and during the lives of elders who had seen God's work for Israel. It points to a living memory of the conquest, rather than a story invented long afterwards.
And Joshua saved Rahab the harlot alive, and her father's household, and all that she had; and she dwelleth in Israel even unto this day; because she hid the messengers, which Joshua sent to spy out Jericho.
Why Christians suggest different dates
Some notes in Joshua look back on events from a later point, which may mean they were added after the main events. For example, Joshua 15:63 says Jebusites were still living alongside Judah in Jerusalem. Later, David took Zion, the stronghold associated with Jerusalem, and established his rule there (2 Samuel 5:6-9).
Some readers take this as evidence that this note was written before David captured the city. Others think such descriptions may reflect a longer, more complex history of peoples living in the area. Faithful Christians therefore differ on whether the final editing happened soon after Joshua's death or later in Israel's history.
As for the Jebusites the inhabitants of Jerusalem, the children of Judah could not drive them out: but the Jebusites dwell with the children of Judah at Jerusalem unto this day.
What can be said with confidence
The careful answer is that Joshua contains material from Joshua's lifetime and was completed in its final form after some of the events it describes. Scripture does not name a final editor or give a calendar date, so it is wise not to claim more than the text does.
Its central concern is clearer than its publication date. Joshua calls Israel to give God their loyal service and to make a real choice about whom they will serve (Joshua 24:15). Reading the book for that message can be more fruitful than trying to force certainty where the Bible leaves questions open.
And if it seem evil unto you to serve the Lord, choose you this day whom ye will serve; whether the gods which your fathers served that were on the other side of the flood, or the gods of the Amorites, in whose land ye dwell: but as for me and my house, we will serve the Lord.