It was not immediately clear what caused the explosion in the speedboat but police said it was not caused by a bomb. Indonesia has a poor maritime safety record and there have been similar incidents in the past where no foul play was detected.
Authorities said the woman killed on the boat, which was heading for the nearby holiday island of Gili Trawangan, was a foreigner but that they were verifying her identity before releasing more details.
The 14 injured passengers included nationals from Portugal, Germany, Australia, South Korea and Britain, according to preliminary information from the police.
The boat was carrying 35 passengers, all foreigners, and four crew, and had just left Padang Bai port in eastern Bali on Thursday morning when the blast occurred.
"The explosion happened five minutes after the boat departed," local police chief Sugeng Sudarso told AFP, adding the vessel had been about 200 metres (yards) from the port.
He said the dead woman and injured passengers were being taken to hospitals on the island.
He said authorities were investigating and police and the bomb squad had been deployed, although he later ruled out a bomb as the cause.
"Based on the testimony (from passengers) and from what I saw on the scene, the explosion came from the fuel tank," he said.
"Above it was a battery, maybe there was a short circuit that affected the fuel tank."
The Indonesian archipelago of more than 17,000 islands is heavily dependent on ferry services but the industry has a poor safety record and fatal accidents are common.
Last year, dozens of tourists were injured when small explosions hit a ferry crossing between Bali and the neighbouring holiday island of Lombok.
The explosions were an accident and thought to have come from the fuel tank of the ferry, which was carrying 129 passengers, most of them tourists.
However fears have also been growing in Indonesia that radicals who have headed to fight with the Islamic State (IS) group in the Middle East could encourage supporters back home to launch attacks, or may launch attacks themselves on their return.
In January, a gun and suicide bomb attack claimed by IS in the capital Jakarta left four attackers and four civilians dead.
Bali has been attacked by Islamic radicals before. In 2002, more than 200 people, mostly foreign tourists, were killed in bombings on the island.
A pocket of Hinduism in Muslim-majority Indonesia, Bali attracts millions of foreign visitors every year due to its palm-fringed, tropical beaches and picture-postcard temples.