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) like the case of an apostate that attacks Islam  
- Life (nafs) like somebody who kills an innocent  
- Lineage / Progeny (nasl) like zina  
- Intellect (‘aql) like drinking/selling alcohol  
- Property / Wealth (mal) 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.