This is a pure play. By that I mean there is a precise (?) U.S. equivalent, bartender, so that the use of barman in the U.S. can be explained only by the desire to use a Britishism, or the conscious or unconscious imitation of others who have done so. (I suppose another possibility is retrograde sexism.)
According to this Google Ngram, the use of barman increased about 20 percent in the U.S. between 2000 and 2008 (actually, between 2000 and 2005; it has held steady since then):
(It’s interesting to look at an Ngram, below, showing the use of barman [blue] and bartender [red] in British English between 1920 and 2008. At least since about 1960, it appears to be a case of an encroaching Americanism, with the two terms recently nearing equality. However, barman is still used about twice as often in Britain as in the U.S., and bartender is six or seven times more prevalent in the U.S.
)
The New Yorker has used barman 34 times from 1937 to the present, including in a 1939 poem called “Forsaken Barman,” a 1964 Talk of the Town piece called “Barman,” and this, from a January 30, 2012, Profile by Nick Paumgarten:
He cuts off the drinks, keeps spare umbrellas on hand for sudden squalls, shuffles customers around to make space for someone’s mom, and, like any barman with a following, dispenses a lot of free drinks.
But you can find the word in all sorts of other sources as well:
“Whiskey You’re the Devil”: The best version of this traditional song is the one you sing right before the barman kicks you out. (Rosie Schaap, New York Times, March 8, 2012)
I grabbed a spot at the bar and was immediately greeted by the friendly barman, Christopher. (HoustonPress.com, March 2, 2012)
Etc.
notoneoffbritishisms.com 
Bartender of course is gender neutral, but in Britain we not only have barmen but barmaids.
Also a few bar stewards.
Now you have prompted me to investigate “barkeep,” the term I use primarily, as “barman” is largely unfamiliar in my experience, found almost exclusively in print, not spoken, and “bartender” is awkward on the tongue, having too many syllables to call out over the din of the bar.
When one asks Google to “define barkeep”, it replies, “bartender: an employee who mixes and serves alcoholic drinks at a bar”. Apparently, the term is derived from “barkeeper: a person who owns or manages a bar where alcoholic beverages are sold.”
“Barkeeper” is found in both U.S. and U.K. literature as far back as Google’s Ngram Viewer searches, i.e., circa 1800, but mostly in the U.S., apparently, as suggested by the U.K.’s extremely jagged history. Overall, its use rose gradually, reaching its peak(s) around the turn of the last century and during WW I, then began a precipitous decline during Prohibition, and has continued a more gradual decline ever since.
“Barkeep,” on the other hand, (inferring again from Ngrams) began its popularity after the American Civil War, and driven primarily by the Yanks, reached its peak during WW II. Its use hit a lull during the “70s, but has risen again in popularity, but not to the point of its heyday, when I was beginning my education.
A barkeeper often kept a barmaid.
But the maid went the way of the waitress and the stewardess.
I give up. “Flight attendants” replaced “stewardesses”; what/who replaced “waitresses”? Were “waiters” replaced, too? (I do know that we still have “actresses,” while “actors” have become gender-neutral.)
Since you brought it up, I’ve noticed a lot less “waiters” and “waitresses” and a lot more “servers.” I don’t know if this is because the waitstaff prefer to “serve paople rather than “wait on” them, or if it’s just corporate America’s attempted to make things more gender neutral. I was curious, though, to know what the trend is in the UK, and I would have thought that “barman” and “barmaid” would have gone the way of the “stewardess.”
I’m fairly sure that Randy Newman wasn’t trying to sound British when he used the word “barman” in his song “Guilty” about 40 years ago.
True–probably just realized that “Got some whiskey from the bartender” didn’t scan.