The title question is better phrased as: when did 'analytic philosophy,' with something very close to its current meaning, become an actor's category?
I started thinking about this after reading Ryle's 1929 review of Heidegger's Being and Time in Mind; interestingly, it did not really contain any of the things so-called 'analytic' philosophers are 'supposed to' say about so-called 'continental philosophers.' Ryle does not treat Heidegger as somehow alien, or as engaged in a fundamentally different pursuit.
Anyway, here's a google books ngram, from 1900 to 2000, with 'analytic philosophy' in blue and 'continental philosophy' in red.
Here's a bigger version of the graph. You'll see that there's no real significant appearance of 'analytic philosophy' until the early 1940s, and that 'contintental philosophy' doesn't appear with much frequency until much later.
With a little bit of googling, I found John Wisdom's 1934 Problems of Mind and Matter referring to analytic philosophy as a definite type of philosophy. However, 'analytic philosophy' in that book appears to be more narrowly confined to something like G.E. Moore's analysis. For example, Wisdom says in the introduction: "Speculating and analyzing are operations which differ in kind: the object of the one is the truth; the object of the other is clarity. It is with the latter that we shall be concerned. ... [T]he analytic philosopher... is not one who learns new truths, but one who gains new insight into old truths" (1-2). Although this characterization does capture an important part of analytic philosophy, I think it leaves out a large amount of what we today think of as analytic philosophy. I have not read through the whole book yet, so I could be wrong about Wisdom restricting his meaning to Moorean analysis. I also found a 1935 Analysis article by A.C. Ewing, "Two Kinds of Analysis," in which the phrases "the analytic school" and "analytic philosophy" apparently apply to Moorean analysis and its adherents (Russell's analysis of descriptions is also mentioned as an example).
In a 2-part 1936 Journal of Philosophy essay, Ernest Nagel reports on a Bildungsreise he took in Europe. The title of the essay is "Impressions and Appraisals of Analytic Philosophy in Europe (I, II)." Here Nagel unites under the heading of 'analytic philosophy' Moorean analysis, early and middle Wittgenstein, the Vienna Circle and their intellectual allies, and the Polish logicians and nominalists. This is the first instance I could find via quick googling of 'analytic philosophy' meaning roughly what it does for us today. But my search has been very casual and cursory; I expect a more careful and thorough investigation will turn up earlier uses of 'analytic philosophy' in roughly our sense. If you find one, please post it in the comments.