local $/="\$\n";
open(A, 'file');
while(<A>) {
   if (m/^(\w+)\n/) { #use whatever reges matches 'DATA' here
      open(my $fh, ">$1") or die "Couldn't open '$1' for write: $!";
      print $fh $_;
      close $fh;
   }
}
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Add in some error checking, and you should be golden. This method has the advantage of only holding one chunk of data in memory at a time, so it scales well. Beware however that this will happily clobber files if there are two secions with the same header. To avoid that, open the output file for append ('>>' instead of '>').