It's not so easy to trace the origin of the Russian word 'banya' (this is the spelling provided by the dictionaries, if they ever have such an entry) The most popular version is that the word is borrowed from Greek together with 'angel', 'icon', 'mathematics', 'torch', 'bed' etc.
In the 4th Ode of the Canon on Pentecost St. John of Damascus (the 8th century) wrote:
"Having by a word mixed the divine bath of rebirth
With a compounded nature,
You rain down upon me a stream from your immaculate,
Pierced side, O Word of God,
Sealing me with the fervour of the Spirit"
The original text was rendered from Old Greek into Old Church Slavonic where the underlined phrase was translated into 'bania pakibytia' in the meaning of 'purification for eternal life' (this English version is another translation from Slavonic)

Another variant is that the word was borrowed from Turkish 'hamam' - a Turkish hot-steam bath

Still another variant - from Latin's balneum (as well as French 'bain' and English 'bath')

So, one may choose the version he likes most

The only thing worth mentioning is that I failed to find the word 'bania' in various dictionaries of the English language.