The caption accompanying the photo read: 'Pitza on bed lv moo x'.
Bill Billingham is the brother of SAS hero Mark Billingham – a star of British TV show SAS Who Dares Wins and a former private body guard for Brad Pitt, Angelina Jolie and Tom Cruise.
Bill Billingham was understood to have been taken to hospital with stab wounds to the stomach, and is set to be questioned in due course.
Police were called to the address at Brownhills, near Walsall, UK, at 9.15pm on Saturday 20 January where the young girl was found with serious injuries.
She was taken to hospital but died a short time later.
Detective Inspector Jim Colclough, from the homicide unit, said: ‘We are treating this as a domestic incident and are not looking for anyone else in connection with the child's death.
‘Our family liaison team are supporting the family of the little girl, who are naturally devastated by her death.
‘Our thoughts remain with them.’
Beth Taundry, a relative of the little girl, took to social media to say: 'Don't bombard anyone with messages, I know that people will want to pass on their condolences.
'We are far too hurt and far too confused right now to deal with logging onto social media and seeing some of the gossip that people in this area are writing.
'Although on behalf of our family I do want to thank everybody for their concern and the general messages hoping all is okay.
'The least everyone can do is keep us in your thoughts.'
Another family member said: 'We have only just come back home after finding out the news.
'We have hardly slept and are just trying to understand and process what has happened.
'We still don’t know exactly what has took place at the this point we are truly devastated.'