Is there a book of this series?
Yes. The first two episodes and the background story of Series One (UK, 2010) were based on Chris Ryan's book "Strike Back" (published in 2006), which is set in Lebanon rather than Iraq.
I am confused. Is this a British or an American show? And who is Richard Armitage?
Series One of this show was produced in 2010 by Left Bank Pictures for Sky in the UK. Richard Armitage was the lead, Sgt. John Porter. Series One was not shown in the USA until late 2013 when it was aired under the name "Strike Back: Origins."
Series Two and the following ones were produced from 2011 by British Sky Television for Sky TV in the UK and Cinemax in the USA. They star Philip Winchester and Sullivan Stapleton as Stonebridge and Scott, Rhona Mitra as Major Rachel Dalton is their commanding officer.