Spitting image. I've never heard the other.

It's interesting.  I do not think I have ever heard 'splitting image'.  However, 'spitting image' is a term often used in expressions here in the USA, pronounced 'spittin' image' here in the Ozarks:

"He is a spittin' image of his father."

I agree with the others that 'splitting image' isn't 'right'. (Or as 'right' as anything is right or wrong when it comes to idiomatic language! ). It makes sense, though - as if the image had been split to produce two identical forms. If anything, it makes more sense than the 'correct' version.

The versions most heard these days are 'spitting image', 'spittin' image, and 'spit and image'. And interestingly, the one closest to the original is in fact 'spittin'.

The idiom, I believe, comes from an ancient creation myth that beings were created by being spat out of the mouths of gods. An ancient forms of cloning, if you like. A variant form of the past participle was, at one time, 'spitten', so the expression started out as 'spitten image'. It then became corrupted, mostly to 'spitting, as the form 'spitten' fell out of use.

P.S. The Oxford Dictionary reference spells it out as "spitting image," but all I can say is that I never have heard it spoken--until I saw that web page--spelled out in print with the "g." I have always seen and heard "spittin' image."

I am sorry to say "spittin' image" is correct, and "splitting image" is incorrect.

Before I look up the dictionary entries other have posted, let me say that I think "spit" here is a contraction of "spirit," and that "spittin' image" is a contraction of "spirit and image." Now let's see if I was right. Well, it seems I am wrong: "the idea of a person apparently being formed from the spit of another."