A BJP worker allegedly killed himself after he was threatened by TMC supporters, police added, citing a complaint filed by his family.
Defying a ban on gatherings of more than four people, hundreds of Banerjee supporters clashed with BJP rivals outside polling stations in Nandigram, where Bannerjee is contesting her seat.
Both parties accused each other of staging raids on local offices and the homes of party officials and supporters.
One group attacked a television channel truck, raining down bricks, rocks and iron bars on the vehicle before the journalists escaped.
"Voters are being intimidated," one national Election Commission official told AFP, speaking on condition of anonymity. "Many are too scared to leave their homes."
Police staged baton charges to disperse gatherings outside polling stations in several towns, the official added.
Despite the violence, thousands still queued outside polling stations to cast votes.
The West Bengal polls are being held over eight days until April 29, with Thursday's second phase involving 30 constituencies.
The results will be announced on May 2, alongside several other polls in Assam, Kerala, Tamil Nadu and Puducherry.
The BJP has spent years preparing to mount its biggest challenge yet in West Bengal, home to 90 million people, and expand its state-level power beyond its Hindi-speaking northern heartlands.
The party won landslides in national elections in 2014 and 2019, but much of the southern half of India remains outside its local control.
Modi has condemned corruption in West Bengal while Banerjee, who has campaigned in a wheelchair after an accident in her car last month, has accused the prime minister of seeking to impose a hostile rule by "outsiders".
The second of three phases of polling in the northeastern state of Assam -- where the BJP is looking to hold on to power -- was also held on Thursday.