ITV's Broadchurch is close behind with five nominations, while Channel 4's conspiracy thriller Utopia has four.
The awards honour behind-the-scenes programme-makers and will be handed out on 27 April in London.
Peaky Blinders creator Steven Knight said: "It's incredibly gratifying when truly creative people who are truly committed... are rewarded."
James Corden and Mathew Baynton received their first writing nominations for BBC Two's The Wrong Mans.
They are up against Steve Delaney and Graham Linehan for BBC Two's Count Arthur Strong; Sam Bain, Jesse Armstrong and Tom Basden for Channel 4's Fresh Meat; and Graham Linehan, who gets a second nod for Channel 4's The IT Crowd.
The best drama writer contenders are Chris Chibnall for crime drama Broadchurch; Dominic Mitchell for supernatural BBC Three series In the Flesh; Sally Wainwright for BBC One's romantic drama Last Tango in Halifax; and Dennis Kelly for Utopia.
Documentaries Educating Yorkshire (Channel 4), Her Majesty's Prison Aylesbury (ITV), The Murder Trial (Channel 4) and The Unspeakable Crime: Rape (BBC One) are all recognised in the best factual director category.
The best director in fiction nominees are James Strong (Broadchurch), Otto Bathurst (Peaky Blinders), Jane Campion and Garth Davis (BBC Two's Top of the Lake) and Marc Munden (Utopia).
ITV's Downton Abbey received a nomination for best costume design, alongside BBC Worldwide's and Starz's Da Vinci's Dragons, BBC Two's An Adventure in Space and Time and ITV's The Suspicions of Mr Whicher: The Murder of Angel Lane.
It was announced last week that the crew behind BBC One's Strictly Come Dancing will receive a special Bafta Craft Award in recognition of their achievements over the last 10 years.
Bafta chairman Andrew Newman said the show had "set new standards for talent and creativity". Actor Stephen Mangan will host the ceremony.
BBC Two gangster drama Peaky Blinders has landed six nominations for the Bafta Television Craft Awards.