I have searched but I did not find any hadeeth that the prophet, peace be upon him, used to recite Surat Al-Asr before leaving a gathering. Is it an innovation?.
Abu Dawood (4859) narrated that Abu Barzah al-Aslami (may Allaah be pleased with him) said: The Messenger of Allaah (blessings and peace of Allaah be upon him) used to say, when he wanted to leave a gathering: “Glory and praise be to You, O Allaah, there is no god but You, I seek Your forgiveness and I repent to You.” A man said: O Messenger of Allaah, you are saying something that you did not say before. He said: “It is expiation for anything that happened in the gathering.”
It was classed as saheeh by al-Albaani in Saheeh Abi Dawood.
The knowledge and virtue of the Sahaabah (may Allaah be pleased with them) and the fact that they so closely followed the Prophet (blessings and peace of Allaah be upon him) indicate that whoever among them knew this hadeeth acted upon it, and we do not need to prove that from each one of them. Rather the basic principle concerning them is that they acted upon the hadeeth and followed the Prophet (blessings and peace of Allaah be upon him).
With regard to reciting Soorat al-‘Asr before leaving a gathering, there is a report which indicates that this was the practice of the Sahaabah (the Allaah be pleased with them).
It was narrated that Abu Madeenah al-Daarimi said: When two men among the companions of the Prophet (blessings and peace of Allaah be upon him) met, they did not part until one of them recited to the other: “By Al‑‘Asr (the time). 2. Verily, man is in loss” [i.e., Soorat al-‘Asr], then one would say salaam to the other. Narrated by Abu Dawood (no. 417); al-Tabaraani in al-Mu’jam al-Awsat (5/215); al-Bayhaqi in Shu’ab al-Eemaan (6/501), via Hammaad ibn Salamah, from Thaabit al-Banaani, from Abu Madeenah al-Daarimi. Classed as saheeh by al-Albaani in al-Silsilah al-Saheehah (2648). He commented on it by saying:
From this hadeeth we learn two things that were among the actions of all the righteous predecessors (may Allaah be pleased with them all):
The first is to say salaam when parting;
Another thing we learn is that the Sahaabah adhered to this practise, namely reciting Soorat al-‘Asr, because they are the least likely of any people to introduce into the religion an act of worship by means of which they seek to draw close to Allaah, except on the basis of what they learned from the Messenger of Allaah (blessings and peace of Allaah be upon him) in word or in deed or by his approval. How could it be otherwise, when Allaah, may He be blessed and exalted, praised them in the highest terms, saying (interpretation of the meaning): “And the foremost to embrace Islam of the Muhaajiroon and the Ansaar and also those who followed them exactly (in Faith). Allaah is well-pleased with them as they are well-pleased with Him. He has prepared for them Gardens under which rivers flow (Paradise), to dwell therein forever. That is the supreme success” [al-Tawbah 9:100]?
And Allaah knows best.