I can work around the issue by making use of nbsp:

use warnings;
use strict;

use OpenOffice::OODoc;
use File::Copy;
use Encode;

my ($i, $my_text);

my @Fnames = ('test.php'); # files to read into document

copy "test.odt.tmpl", "test.odt";
my $doc_out = odfDocument(file => 'test.odt'); # setup output

# ====== add files into output document ======
foreach(@Fnames) # process each php file
{
   open(my $file, "<", $_) or die "Error reading $_: $!";
   $my_text = ""; # init

   for (<$file>)
   {
      # \x{A0} is &nbsp;
      s/^( +)/"\x{A0}"x(length($1))/e;
      $my_text .= $_;
   }

   $doc_out -> appendParagraph(text => encode('UTF-8',$my_text),
                               style => "Default"); # write a full php
+ file to odt
}

$doc_out -> save; # done
[download]

I also made a few perly and safety changes. The heart of the solution is the s/// and the encode(). My openoffice draws the non-breaking spaces with a different background color which is a bit distracting. I'm not sure if that can be worked around.