Hanukkah commemorates the rededication of the Temple in 164 BCE after the Maccabean revolt against Seleucid rule, and the earliest accounts of it do not mention the oil.
First Maccabees and Second Maccabees, written within roughly a century of the events, describe the desecration of the Temple by Antiochus IV, the revolt led by Mattathias and his sons, the recapture of Jerusalem, the cleansing of the sanctuary and the institution of an eight-day festival. Second Maccabees explicitly models the eight days on Sukkot, which the rebels had been unable to keep in the field, and Sukkot is eight days. Josephus, writing in the first century, calls it the Festival of Lights and says he does not know why that name was given, which is a remarkable admission from someone that close to it.
The story of a single cruse of undefiled oil that burned for eight days appears first in the Babylonian Talmud, at Shabbat 21b, compiled centuries later. It is the story every Jewish child learns, and it is not in the historical sources.
This is not a case of one account being false. It is a case of a festival whose commemorative content shifted. The Maccabean books celebrate a military and political victory won by a priestly family that then founded a dynasty; the rabbinic tradition, writing after two catastrophic revolts against Rome, has notably little enthusiasm for armed rebellion, and the miracle relocates the festival's meaning from the sword to divine provision.
Neither book of Maccabees is in the Jewish canon. Both are in Catholic and Orthodox Bibles.