Most lists of Dashavatar include Buddha as the ninth avatar of Vishnu. Some include Balarama, the elder brother of Krishna, as the ninth avatar instead of Buddha. I have a problem with both.
Balarama was indisputably the avatar of Sesha Nag, the serpent on which Vishnu reclined in the lake of Kshirsagar, just as Laxman was the avatar of Sesha Nag during Rama avatar period. Hence to include Balarama as an avatar of Vishnu is anomalous.
The teachings of Buddha are quite divergent from Hinduism. If you ask any Buddhist, whether a follower of any ancient school of Buddhism or the neo-Buddhists who have converted from Hinduism after India’s independence, if he is a Hindu then you will be met with disbelief or contempt. No Buddhist considers himself or herself to be a Hindu. Yet there are many Hindus who consider Buddhism an offshoot of Hinduism – ridiculous but true.
The inclusion of Buddha as the ninth avatar of Vishnu was nothing but an unsuccessful attempt to prevent the massive conversions from Hinduism that were accompanying the sermons of Buddha. In fact Hinduism was perhaps at its worst at the time Buddha started preaching. Social exploitation in the name of religion was beginning to entrench itself and many of the Brahmin class were corrupt and greedy. No wonder there was disillusionment and Buddha found a ready following.
The reconciliation of the differences between Buddhism and Hinduism is even more amazing. The standard line followed in most scriptural texts is that Vishnu as Buddha intentionally preached false scriptures so as to prevent the demons from accessing the true faith and therefore preventing their salvation. If some humans were fooled into believing Buddha then it was an inevitable side effect.