1. Set a flag everytime a header starts.
2. Grab everything for that chunk and write it to its city's file.txt.
3. Reset the flag when the next header is encountered

#!/usr/bin/perl

use strict;
my ($file);

open(FILE, "<file.txt");
while(<FILE>) {
    chomp($_);
    if(/CITY/ ... /FLORIDA/) {
        if( ($file) = $_ =~ m/.+CITY:\s+(\w+)\s+/ ) {
           close(FH);
           $file .= '.txt';
           open(FH, ">>$file");
        }
        print FH "$_\n" unless $_ =~ m/FLORIDA/;
    }
}
close(FILE);
[download]

Only note that if the file is big, and there is not a lot of different files to create, calling open everytime a file must be update can be a considerable overhead (I've been in the same situation just a few days before).

If the different files can be determinated before the loop, you can open all of them and the use the apropiate (i.e., keeping them in a hash). When they're not known until runtime (it was my case), I used a closure returning lexical filehandles from a hash. Something like that:

{
        my %handles; # Closure keeps the hash between function calls
        sub handle{
                my $city = shift;
                my $temp;
                unless ($handles{$city}){
                        open $temp, ">", "$city.txt"
                                or die "$!";
                        $handles{$city} = $temp;
                }
                return $handles{$city};
        }
}
[download]

It worked for me and reduced the execution time about a 50%. Hope it helps.