The child was admitted to Christchurch Hospital on New Year's Eve with serious injuries.
A scene examination is ongoing at a residential property in Cuffs Road in Wainoni.
Detective Inspector Michael Ford said residents should expect to see a continued police presence in the coming days.
He said an incident involving a young and vulnerable child was "extremely concerning and unsettling for the community".
He thanked them for their support and cooperation in the investigation.
Police continue to appeal for further information as to how the baby was injured.
Anyone with information is asked to contact Christchurch police on 105. Information can also be provided anonymously via Crime Stoppers on 0800 555 111.