This is one of those cases where for readability sake, I would separate the split logic from the case logic. This isn't going to win any perl golf competitions but future maintainers will not try to hunt you down!

#!/usr/bin/env perl

use strict;
use warnings;

my $input = "I am a GIRL";
my $output = [];

# split input by non-words ... if you need to keep non-word 
# characters then you'll need a different regex
foreach my $word ( split( /\W+/, $input ) ) {
  # if the word is all upper case and it's not a single letter
  if( $word =~ /^\p{Uppercase}+$/ && length( $word ) > 1 ) {
    push( @$output, $word );
  } else {
    push( @$output, lc $word );
  }
}

print join( ' ', @$output ), "\n";
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update: Fix regex from /\p{Lu}+/ to /^\p{Uppercase}+$/