The boat the baby was traveling on capsized shortly before being intercepted by the Italian coastguard, ANSA and other Italian news agencies reported on Wednesday.
All other passengers were rescued, including the baby's mother, a teenager from Guinea, ANSA said. It said the migrants had set off from the Tunisia city of Sfax, a well-known launchpad for sea journeys to Europe.
Lampedusa, Italy's southernmost point and a first port of call for people crossing from North Africa, has long been a flashpoint in Europe's migration crisis.
On Tuesday, footage showed queues of flimsy boats, packed with migrants, all waiting to dock at Lampedusa's port as the daily tally of arrivals hit a new high.
Prosecutor Giovanni Di Leo said 112 vessels arrived on Tuesday, carrying more than 5,000 migrants, far surpassing the previous record of 63 boats registered on one day last month. Meanwhile, migrant landings continued, with about 1,300 new arrivals on Wednesday morning.
Di Leo compared the emergency on the small island to that Italy experienced in 1991, when a single ship brought about 20,000 Albanians to the southern city of Bari.
"We are all tired and exhausted both physically and psychologically, the situation is becoming unmanageable and unsustainable," Lampedusa Mayor Filippo Mannino told the Adnkronos news agency on Wednesday.
Migrants who arrive on Lampedusa are regularly transferred to Sicily to ease overcrowding, but when arrivals surge, this cannot happen quickly enough.
The island's reception centre had more than 6,000 people in on Wednesday, the Italian Red Cross said, compared with its official capacity of around 400.
Overall, about 118,500 boat migrants have arrived in Italy since the start of the year, according to interior ministry data last updated on Tuesday morning.
The figure is almost double what was recorded in the same period of 2022.
Prime Minister Giorgia Meloni, elected last year pledging to fight irregular immigration, has failed to bring down the numbers as a crackdown on migrants and economic crisis in Tunisia, as well as continuing chaos in Libya, have led to more sea crossings from North Africa.
Three children and a woman died when their migrant ship sank off Sfax, a judge there told Reuters on Wednesday.