Some time ago, Paris police launched a manhunt for three gunmen.
But, what if there are three women who carry guns, shoot and escape, is it still appropriate to say "manhunt for three gunmen"? Will it have to be "womanhunt for three gunwomen"?
Yes, you would still use the word "manhunt". For example, a review of Thelma & Louise includes this quote:
Ridley Scott’s 1991 road movie Thelma & Louise is often cited as the singular feminist movie of the decade, but in reality it’s so much more than that. Geena Davis plays Thelma, a housewife stuck in a mundane marriage. Sarandon plays Louise, a waitress who always finds herself alone while her musician husband is out on the road. The two decide to take a road trip but soon become fugitive when Louise shoots and kills a man who attempts to rape Thelma. Soon, they find themselves the targets of a massive federal manhunt.
According to Google NGrams, "manhunt" is about 600 times more common than "womanhunt". Similarly, "gunman" is about 600 times more common than "gunwoman". Whereas "armed man" is only about 12 times more common than "armed woman". Google NGrams does not find "female gunman" at all.
Men are much more likely to commit crimes using guns than women, so it is hard to find descriptions of female gunmen. Here is an example of an author using a phrase to avoid choosing between "female gunmen" or "gunwomen":
Man is derived from a term that was gender neutral and this sense persists today in many terms, such as manslaughter, so manhunt is the correct term to use.
If you wish to find a gender neutral alternative to "gunman" and a more specific term such as "armed robber" is not appropriate, the term "shooter" can be used. This is a widely used term(more common even than gunman) and has become much more frequent in recent years. The term "Active Shooter" is defined by the U.S.Department of Homeland Security as ""an individual actively engaged in killing or attempting to kill people in a conﬁned and populated area" and would be appropriate in this instance.
My humble opinion, as I am a not-mother-tongue-English-speaker, is "man" is coming from mankind which designates an human being, male or female. So manhunt will be appropriate regardless the sex of the hunted. I guess the same applies to "gunman" which is the same regardless if the gun bearer is male or female. Try polling a dictionary with the term "gunwoman".
It's perfectly acceptable to use terminology that implies a gender if you attach a male term to something negative or a female term to something positive. It's only sexist when your words imply that something positive must be male or something negative must be female. :-)
In the 1920s the female equivalent of a "gunman" was a "gun moll".
I have never heard anyone say "womanhunt" or "gunwoman". If someone was worried about implying that the person with the gun is male when that person's sex is unknown, they'd probably say, "A search for the suspect" rather than "A personhunt for the guncreature".
The word means "human hunt", not "male hunt". The man- part of the word is meant in the larger generic sense, not the gender-specific sense.
I suppose in certain circumstances it could be used as a humorous double entendre, with the interpretation in your more limited sense furnishing the pun. For example, a group of ladies out for a night on the town might say they are on a manhunt.