Some student of knowledge and many people in our country(Bangladesh) say that Drinking water in 3 breaths is a 'Sunnah' . Are there any authentic hadith or evidence backing this up? Or is this another innovation?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

It was the guidance of the Prophet to drink by three times, and breath each time between them when he removes the vessel away from his honorable mouth.

Anas ibn Maalik narrated that when the Prophet drank, he used to breathe three times in the course of a drink and say: “It is more wholesome, thirst-quenching and healthier.” [Abu Dawood, An-Nasa-i and Al-Bayhaqi]

Moreover, Abu Hurayrah narrated that the Prophet said: “None of you should breathe into the vessel when he is drinking from it, but when you want to breathe, you should move the vessel away, and then breathe [outside the vessel]” [Al-Hakim and Ath-Thahabi]

Badr Ad-Deen, from the Hanafi School of jurisprudence, said in Sharh al-Bukhari when explaining the Hadeeth, “If one of you drinks, he should not breathe into the vessel”: “If you say, is the ruling limited to drinking water only or does it apply to other drinks as well: I reply: “The prohibition that is mentioned is not limited to drinking water only; rather, the same ruling applies to all other beverages.” [End of quote]

The Prophet used to say Bismil-lah (in the Name of Allah) in each of the three times and he praised Allah.

Abu Hurayrah narrated that the Prophet used to drink by three sips (breaths), when he brings the vessel closer to his mouth, he used to say Bismi Allah, when he takes the vessel away from his mouth, he praised Allah; he would do that three times.” [Al-Khara-iti, Al-Albani: Authentic]

Allah knows best.