For some reason the verb sounds very bad there. I would never say 'a fishing man'. Yet the structure is grammatically sound: a sleeping baby, a crying baby, a falling branch, a dying swan, a burning log. But we wouldn't talk about a skiing man, a playing child: my guess is that the disallowed ones are more or less organized human activities (compare playing with crying), but I can't see why that makes a grammatical difference.
Note that in all these the noun is accented: a crying 'child, a burning 'log. If you use the gerundial noun (the name of activities such as skiing and fishing) you can use that noun as a modifier, but it gets the accent: a 'skiing instructor, a 'fishing expert. (A fishing expert is an expert on fishing, not an expert who is fishing.)