Is one who prolongs the Sunnah of Fajr sinful if the time for the obligatory prayer expires unintentionally?

There is no sin for what occurred unintentionally, as Allah Almighty says: "Our Lord, do not impose blame upon us if we forget or make a mistake," and as the Prophet, peace and blessings be upon him, said: "Indeed, Allah has overlooked for my Ummah their mistakes, their forgetfulness, and what they are coerced into." The Prophet, peace and blessings be upon him, would not prolong the two rak'ahs of Fajr.