The 56-metre (180-foot) long luxury yacht had been moored off Porticello, east of Palermo, when violent winds and rains swept suddenly up the coast before dawn.
Lynch's wife Angela Bacares was among 15 people rescued but the businessman and his daughter Hannah Lynch were missing, Salvo Cocina, head of the Civil Protection Agency in Sicily, told AFP.
Bacares suffered light injuries and was spotted by AFP being wheeled into a hospital in Termini Imerese.
Lynch, 59, is a celebrated technology sector entrepreneur and investor.
Divers who recovered a first body in the area earlier on Monday had seen one of the missing six trapped inside the sunken vessel, according to a source close to the recovery operation.
"It was terrible. The boat was hit by really strong wind and shortly after it went down," a British survivor named as Charlotte Golunski told ANSA news agency.
The Bayesian, flying the British flag, had 22 people aboard, including 10 crew members, the coast guard said.
Most of those aboard were British, and the passengers were guests of Lynch, according to Italian media.
Survivors included two people with joint French and British nationalities, a Sri Lankan, a New Zealander and an Irish national, according to media reports.
'No longer there'
Camper & Nicholsons, the managers of the yacht, said in a statement that they were "assisting with the ongoing search" for the missing people.
Italian authorities have opened a probe into the incident.
The incident comes after Mike Lynch was acquitted on all charges in a San Francisco court in early June after he was accused of the massive fraud linked to the sale of his software firm Autonomy to Hewlett-Packard.
The Bayesian was a luxury superyacht built by the Italian shipbuilding firm Perini Navi in 2008.
It had an 11.51-metre mast and could reach speeds of up to 15.5 knots, according to the Boat International website.
"The boat was all lit up. Around 4:30 am, it was no longer there," said one witness cited by ANSA.
"A beautiful boat where there was a party. A normal joyous vacation day at sea turned into tragedy".
