Two-month-old baby died in a hospital on Saturday, several hours after the family reached Greek island of Agathonisi by boat from Turkey.
There was no information given by authorities about the child’s identity or cause of death.
The baby was amongst the 2,500 refugees who boarded a Greek ferry to reach the port of Piraeus on Saturday.
Thousands of refugees who managed to successfully reach the island, have made a temporary camp and blocked a central road on the island.
"It's difficult, there is shortage in water, we have kids, we have small babies, we have a shortage in all kinds of service," a Syrian refugee who reached the island told Reuters.
Meanwhile, riot police is struggling to handle with the increasing number of refugees who have fled from the Syrian civil war by crossing over from Turkey.
Police used tear gas to disperse thousands of refugees on Saturday night, as they attempted to cross barriers in order to board a ship, a police source said.
If the Greek government does not address the situation quickly, the island will boycott the general election on Sept 20, warned the mayor of Lesbos.
Last month, Former Greek prime minister Alexis Tsipras, resigned and made arrangements for a snap election, which will be held in mid September.
Authorities have estimated that about 2,500 people have died his year, while attempting to seek refuge in Europe.
"With all my heart, I hope that the government that will take over after the elections will tackle the major economic issues along with the migration and refugee issues prudently and with responsibility," he said after meeting with Greek President Prokopis Pavlopoulos and Interim Prime Minister Vassiliki Thanou.