A game with the same board and pieces, Tabula, was played in Rome in the fifth century. It derived from the "Game of Twelve Lines", played around year 0.
From Wikipedia:
Tάβλι (Tavli), i.e. tabula meaning 'table' or 'board' in Byzantine Greek, is the oldest game with rules known to be nearly identical to backgammon. It is described in an epigram of Byzantine Emperor Zeno (AD 476–491).[20]
The board was the same, with 24 points, 12 on each side. Like today, each player had 15 men and used cubical dice with sides numbered one to six. [20]
The τάβλι of Zeno's time is believed to be a direct descendant of the earlier Roman Ludus duodecim scriptorum ('Game of Twelve Lines') with the board's middle row of points removed, and only the two outer rows remaining.[21] Duodecim scriptorum used a board with three rows of 12 points each, with the 15 men being moved in opposing directions by the two players across three rows according to the roll of the three cubical dice.[20][21] Little specific text about the play of Duodecim scriptorum has survived;[24] it may have been related to the older Ancient Greek dice game Kubeia. The earliest known mention of the game is in Ovid's Ars Amatoria ('The Art of Love'), written between 1 BC and 8 AD. In Roman times, this game was also known as Alea, and a likely apocryphal Latin story linked this name, and the game, to a Trojan soldier named Alea.[25][26]
Wikipedia also includes more history of the evolution of tabula from other race games, but that is off topic for the question at hand.
Wikipedia's citations:
20: Austin, Roland G (1934). "Zeno's Game of τάβλη". The Journal of Hellenic Studies. 54 (2): 202–205. doi:10.2307/626864. JSTOR 626864.
21: Austin, Roland G. (February 1935). "Roman Board Games. II". Greece & Rome. 4 (11): 76–82. doi:10.1017/s0017383500003119.
24: Austin, Roland G. (October 1934). "Roman Board Games. I". Greece & Rome. 4 (10): 24–34. doi:10.1017/s0017383500002941.
25: Finkel, Irving L. "Ancient board games in perspective." British Museum Colloquium. 2007. p. 224
26: Jacoby, Oswald, and John R. Crawford. The backgammon book. Viking Pr, 1976.