open BIG, 'big_file';
my $i = 1;

while (<BIG>) {
  open OUT, ">outfile_" .$i++; #BUG: see follow up below.

  if ( m/^\$\n$/ ) {
    close OUT;
    open OUT, ">outfile_" .$i++;
    next;
  }

  print OUT;
}

close OUT;