From reading copy on behalf of a client, I questioned their use of anytime vs any time. I thought it should be any time (two words.)
What I learned is two-fold: Except in very clear cases such as "I don't have any time to do this," there's no one right answer for which to use. This grammar website I went to said that when in doubt, make it two words.
The second half of what I learned is that "anytime" is a new word. The website says:
Anytime is a new word. Merriam-Webster’s dictionary lists 1926 as the year of its first known use (though earlier instances are easily found in historical Google searches). The Oxford English Dictionary doesn’t list it at all. Some sources say it is an Americanism, and while it’s true that the word is especially common in American publications, it is used throughout the English-speaking world.