The Health Department's chief epidemiologist says all four cases occurred among mothers of the Guna Yala indigenous group.
Dr Israel Cedeno said today that two of the four babies had died. The Guna Yala Indians inhabit a stretch of low coast along Panama's Atlantic shore.
In all four cases, either the mother, the baby or both tested positive for Zika.
Panama has had 264 confirmed Zika cases, with 11 in pregnant women.
The mosquito-borne Zika virus can cause microcephaly, a severe birth defect in which babies are born with abnormally small heads.
