All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
After careful research in the books and references available to us, we could not find any account about the Prophet and the Companions in relation to the children's lying and the way to address this problem. We could also not find any account indicating that the Prophet saw such behavior and ignored it. If he would have seen it, we believe that he would not have ignored it. He used to direct young children regarding improper and impermissible actions on their part. This is evidenced by the following authentic ahaadeeth:
Abu Hurayrah narrated that Al-Hasan ibn ʻAli took a date from the dates of charity and put it in his mouth. The Prophet said (to him) in Persian, "Kakh, kakh! (i.e. do not do so) you know that we do not eat from what is given in charity." [Al-Bukhari and Muslim]
ʻUmar ibn Abi Salamah said, "I was a boy under the care of the Messenger of Allaah and my hand used to go around the dish while I was eating. So he said to me, 'O boy! Mention the Name of Allaah and eat with your right hand, and eat of what is nearer to you.'" [Al-Bukhari and Muslim]
Allaah knows best.