One way of doing it would be using a simple state machine. Here's an untested piece of code:

#!/usr/bin/perl

use strict;
use warnings;

my $STATE = "FILE";

my $name;
my $fh;

while (<>) {
    if ($STATE eq "DATA") {
        if ($_ eq "\$\n") {
            close $fh or die "close '$name': $!\n";
            $STATE = "NAME";
            next;
        }
        print $fh $_;
        next;
    }
    elsif ($STATE eq "FILE") {
        chomp;
        $name = $_;
        open $fh => ">", $name or die "open '$name': $!\n";
        $STATE = "DATA";
        next;
    }
    else {
        die "Unknown state '$STATE'.\n";
    }
}

if ($STATE eq "DATA") {
    close $fh or die "close '$name': $!\n";
}

__END__
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Abigail