Mr Cafferkey, 72, from the Acomb area of York, was last seen by a neighbour on Wednesday 30 September 2020.
Officers have been carrying out searches to find him since he was reported missing on Friday 2 October, but he has not yet been located.
His family are very concerned for his welfare, as his disappearance is out of character.
North Yorkshire Police are urging anyone who sees Mr Cafferkey, or knows where he is, to contact them straight away. Please dial 101, select option 1, and quote reference number 12200173603.