I know Purple Rain is an old movie, but I think it’s pretty good. But, in the last part of the movie, Prince is just sitting in the dressing room, I assume trying to come up with the song.
My question, to musicians, or whomever, can a whole song be arranged on the fly like that? I mean, nobody really knows what Prince is doing until he introduces the song. Can a real band just arrange a great song like Purple Rain on the fly like that?
Clearly that didn’t actually happen in the movie. As a Thing, though, yeah, it can happen. If the song or underlying chord progression and key is known/sussed out real time, and the musicians involved have competency and taste*, they can each gauge the basic feel of the song and add their little bit via feel and knowing what key the song is in.
I’ve done it countless times sitting in on rudimentary 3-chord rock songs, including many I didn’t know before I got thrown into the mix 
I couldn’t do it much past the basic 3 and 4-chord progressions, but someone with basic theory certainly could.
*i.e., understanding that Less is More, and you are trying to add to the whole. Jab with a chord or a lead fill for a measure or two, go from there.
The story behind American Woman shows some truly on the spot creativity.
You can’t get more ‘on the fly’ than the creation of Ray Charles’ “What’d I Say”…the man came up with it off the top of his head on stage while he had twelve minutes of showtime left and needed an encore stat(!).
The late, great Sun Ra reportedly wrote new parts for certain instruments for certain songs off the top of his head and would hand sheet music or written notations to people right before they played.
I have a friend that used to jot numbers and squiggles down on paper while he was at his day job. When he came home from work he would show us these things, grab his guitar and play the song he had written. Then he’d grab a bass and play the bass line. It frustrated him that he couldn’t just play the drum part too so he taught himself to play drums. And yeah, his tiny pieces of paper with squiggles were all the parts to a song, all of which he wrote in his head while he was workings day job.
So yeah, people do write things on the fly.
Not being a fan of Prince, I am ignorant about Purple Rain in movie or song form. But, yes, as WordMan says, good musicians can, under some circumstances, make an on-the-fly head arrangement sound like it was rehearsed.
The most obvious example is a 12-bar blues. Just yell out “blues in G!” and give a countdown, and you should be able to get a respectable tune out of any band. There are a few dozen riffs, intros and endings everyone knows, and good musicians can feel where a tune is going.
Listen to Part One of a Labor Day concert I just recorded. Notice especially Rory Hoffman, who is just doing “what comes naturally” much of the time. (Rory is the guy sitting down, playing mandolin, guitar, clarinet, and accordion, but not all at once.) Sure, this group had some rehearsal – how much, I don’t know, but I suspect it was only a run-thru for some tunes. The rest was arranged on-the-fly.
Also notice some of the cues given by others in the band, like, “Take it, Rory!”, and naturally, he does (only Dick Smothers would say, “no”. 
) The next time they perform that tune, someone else might get the hint for a solo, and the end product changes.
If you are interested, here are parts two & three of the same concert. (I don’t make any money if you click, but it’s nice to have people watch!)
Oh shit! You video’d Rory Hoffman? Man, how cool was that?!
Note: Guitar Player - Players he’s been blind since birth and taught himself to play guitars face up on his lap - Jeff Healey style for those who know him. But this guy’s much more technical. Just amazing to watch and listen to.
Back to the OP’s ? - so this guy Hoffman is one of the most skilled listeners and players around. I look forward to watching more and see how natural his “improv arrangement” playing is. I suspect his work will look seamless to mortals like us.
Maybe I’m remembering the movie wrong, but I thought Wendy and Lisa provided the tune to “Purple Rain” to the Kid on a demo tape, so they would have already known it. Presumably the rhythm section played on it too. So the band would have run through it at least once before the Kid springs it on the crowd.
I’m not a musician and I haven’t seen Purple Rain in quite a while, but I assumed that Prince was just getting himself properly psyched up before going on stage.
No, you aren’t wrong. I believe part of The Kid’s growing up included letting others into his creative process and, more generally, to let his guard down. I always wondered if Prince himself ever learned this lesson.
Wendy and Lisa wrote the music to the song, but in the movie, they got in trouble for practicing at band rehearsals. Prince didn’t even want to hear it. He did have the demo tape though, and the movie shows him listening to it, and then playing in on the piano.
At this point, there were no lyrics or lead guitar part or anything. He seemed to make it up the night before and in the dressing room while he was just sitting there not saying anything before the show. I’m pretty sure the band didn’t even know the lyrics at that point.
But, some actual musicians here say it’s possible, so that’s good enough for me