Is it proven that some children gave allegiance (bay'ah) to the Prophet ﷺ?

Yes, a group of discerning children from among the Companions who had reached the age of maturity (discernment) gave allegiance to him. Asma bint Abi Bakr (may Allah be pleased with her) said: 'Abdullah ibn al-Zubayr came when he was seven or eight years old to give allegiance to the Messenger of Allah (peace be upon him). Al-Zubayr had ordered him to do so. The Messenger of Allah (peace be upon him) smiled when he saw him coming toward him, and then he accepted his allegiance.' (Muslim).