Police launched an investigation following the death of the baby in hospital on March 5th.
The two people arrested have been released on bail.
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A police spokesman said: “We were called around 1-20pm on March 1st to an address in Hapton, following reports a baby had suffered a medical episode.
“Emergency services attended and the child was taken to Manchester Children’s Hospital for treatment, however he sadly passed away on March 5th. His family are now being supported by officers.
“A 60-year-old woman and a 77-year-old man, from Burnley, were arrested on suspicion of section 18 assault and have since been released on bail pending further enquiries.
“An investigation is under way to establish what caused the child’s death. We are keeping an open mind and would ask the public not to speculate.
“Anybody with information is asked to call us on 101, quoting log 614 of March 1st, 2022.”