No there are only hadd punishment for what goes against one of the five fundamental goals of shari'a مقاصد الشريعة which are (I'll quote example cases where the punishment may apply for each of them):
The preservation of ...
- Religion / Faith (din الدين) like the case of an apostate that attacks Islam
- Life (nafs النفس) like somebody who kills an innocent
- Lineage / Progeny (nasl النسل) like zina
- Intellect (‘aql العقل) like drinking/selling alcohol
- Property / Wealth (mal المال) like stealing
And backbiting doesn't go against these and therefore is not punishable. Also note if there was a hadd punishment for that it would surely have been mentioned in the qur'an. Only details (for example: special cases) may appear in the sunnah.
Now what is the punishment for ghaybah or gheebah الغيبة?
The defintion of backbiting can be found in the sunnah like in the hadith saying:
Do you know what is backbiting? They (the Companions) said: Allah and His Messenger know best. Thereupon he (the Holy Prophet) said: Backbiting implies your talking about your brother in a manner which he does not like. It was said to him: What is your opinion about this that if I actually find (that failing) in my brother which I made a mention of? He said: If (that failing) is actually found (in him) what you assert, you in fact backbited him, and if that is not in him it is a slander. (sahih Muslim)
This makes clear that backbiting is clearly one of the major sins.
And the prophet () has seen in his m'iraj the punishment of backbiters:
The Prophet (ﷺ) said: When I was taken up to heaven I passed by people who had nails of copper and were scratching their faces and their breasts. I said: Who are these people, Gabriel? He replied: They are those who were given to back biting and who aspersed people's honour.
Abu Dawud said: Yahya b. 'Uthman has also transmitted it from Baqiyyah, there is no mention of Anas in it. (sunan abi Dawod)
This is the punishment in the hereafter.
The quran also gave an example of backbiting:
... Would one of you like to eat the flesh of his brother when dead? You would detest it. And fear Allah ; indeed, Allah is Accepting of repentance and Merciful. (49:12)
Now even if I said that there's no hadd punishment for gheebah this doesn't mean one cannot punish backbiters in a shari'a conform way in this life. As in cases you may take a backbiter to a court and a qadi (judge) may speak out a verdict which may be some kind of punishment but that would be based on ta'azir (which depends on the case and the interpretation of the qadi).