Thank you all for your insights and shared code. I have not replied since because I am still struggling with this. I have solved the first part: fetching a message from server, thanks to your input. I do something like this:

use Mail::IMAPClient;
use Email::MIME;
use Data::Dumper;

my $imap = Mail::IMAPClient->new(
  Server   => 'abc.com',
  User     => 'xxx',
  Password => 'xxx',
  Ssl      => 1,
  Uid      => 1,
#  Starttls => 1,
);
die "failed to instantiate." unless defined $imap;

$imap->connect or die "Could not connect: $@\n";

my $folders = $imap->folders
  or die "List folders error: ", $imap->LastError, "\n";
print "Folders: @$folders\n";

$imap->select( 'INBOX' )
  or die "Select 'INBOX' error: ", $imap->LastError, "\n";

my $list = $client->search('SUBJECT', 'a new email');
for my $msgid (@$list){
    my $from  = $client->get_header( $msgid, "From" );
    my $subj  = $client->get_header( $msgid, "Subject" );
        my $bsdat = $client->fetch( $msgid, "bodystructure" );
        my $bss   = $client->body_string($msgid);
        my $parser = MIME::Parser->new();
        $parser->output_to_core(0);
        # this saves message IN ONE BIG FILE, text+attachments togethe
+r!!!
        # and the extension is '.txt'!!!!
        $parser->extract_nested_messages(1);
        $parser->output_under('./out');
        my $entity = $parser->parse_data($bss);
        # $entity->parts does not give me the parts 
        # even if message is 'Content-type: MULTIPART/mixed'
}
[download]

I am still struggling with the 2nd part: unwrap a message to local disk, each attachment on its own file. And I am looking for a way to do that seemingly simple and solved-by-now problem either by MIME::Parser or some other package. Alas the prospects look bleak.

The above was put together with code from NERDVANA, Discipulus, talexb !

p.s. edit: my bandwidth is very limited so in order to test this I have setup a minimal mail server (dovecot) in my linux box without the ability to smtp or ssl (to keep things simple). I have used thunderbird in order to copy my multipart test email from a "real" email account's INBOX to the localhost dummy (using 'copy to' in thunderbird) and now I can do the testing without using the net or bothering my MailSP. Of course I could have just saved the email into a file and read from there ...