The baby died several hours after the family landed on Agathonisi, having sailed from Turkey. The infant was treated at a local clinic Saturday but its medics had only limited facilities, AFP cited mayor Evangelos Kottoros as telling public TV network ERT.
Authorities did not immediately provide any further information on the child's identity or the cause of death.
Meanwhile on Lesbos, riot police used tear gas on Saturday night to drive back thousands of refugees and migrants who had tried to force through barriers in a desperate bid to get onboard a ship specially chartered to transfer some of them to Athens on the mainland, a police source said.
A refugee was injured in the clashes and taken to hospital, the source said.
Earlier in the day, hundreds of refugees and migrants staged a protest against the delayed registration process that has frustrated their bid to continue travelling north to other destinations in Europe.
"Asylum, Asylum!" they shouted in front of the island's main refugee centre, with some hurling stones at police.
Several Greek islands are struggling to cope as thousands of people arrive from Turkey on flimsy boats, most of them refugees fleeing war-torn Syria.
In recent days, several islands have seen clashes between police and migrants, and between migrants of different nationalities.
Greece has had more than 230,000 people landing on its shores this year, the vast majority seeking to travel on other EU countries such as Germany and Sweden.