Mr Black, 52, was shot as he drove to work on a motorway in Co Armagh on Nov 1. Dissident republicans claimed responsibility.
The father of two worked at the high-secu-rity Maghaberry jail in Co Antrim.
A 36-year-old man was arrested in Co Tyrone by the PSNI and taken to Antrim police station for questioning.
Meanwhile, in the Republic, a man aged 44 and a woman aged 31 have been detained by gardaí.
The pair were being held under section 30 of the Offences of the State Act 1939 at Ballymun Garda station.
Mr Black was the first prison officer murdered in Northern Ireland for 20 years.