The only suggestion I can make is that in reformatting your information for posting, you have somehow removed your bug. Specifically, running the script:

#! /usr/bin/perl
use strict;
use warnings;
use Archive::Zip qw( :ERROR_CODES :CONSTANTS );

my $zip = Archive::Zip->new();

open (FILE_LIST, '/home/user/d/file_list.txt')  or die "File list open
+ fail: $!";

while (<FILE_LIST>) {
        chomp;
        my $member = $zip->addFile($_);
        print "Add error" unless ($member);
}

close FILE_LIST;

unless ($zip->writeToFileNamed('/home/user/d/backup.zip') == AZ_OK) {
        die "write error";}
[download]

from multiple directories yielded success for me, with input file:

/home/user/a/a.txt
/home/user/b/b.txt
/home/user/c/c.txt
[download]

and where success was measured by the results of the script:

#! /usr/bin/perl
use strict;
use warnings;
use Archive::Zip qw( :ERROR_CODES :CONSTANTS );

my $zip = Archive::Zip->new('backup.zip');
print $zip->memberNames();
[download]

where the only scrubbing I performed was changing user name. The only other suggestion I can make is that a review of the add_file method in Archive::Zip includes the admonition:

NOTE that you shouldn't (generally) use absolute path names in zip member names, as this will cause problems with some zip tools as well as introduce a security hole and make the zip harder to use.