You're not downloading anything in this case, you're simply printing to your local console the output of a remote device (a Cisco network device, apparently). I've reformatted your code below (so it's easier to read/follow), and added a couple of things that will write your output to a file named "output.txt". Note that this code is untested:

use strict;
use warnings;

use Net::Ping;
use Net::SSH2;

my $output_file = 'output.txt';

my $ssh = Net::SSH2->new();
my $username="username";
my $password="password";
my $ip_address = ("ipadderss");
my $ping= Net::Ping->new();

if($ping->ping($ip_address)){
    if($ssh->connect($ip_address)){
        if($ssh->auth_password($username,$password)){
            my $chan = $ssh->channel();

            $chan->blocking(1);
            $chan->shell();
            $chan->write("terminal length 0\n");
            $chan->write("show cdp neighbor \n");

            # open the output file in overwrite mode
            # (use '>>' if you want to append instead)

            open my $wfh, '>', $output_file or die $!;

            while (<$chan>) {
                # instead of printing to the console, print
                # to the file handle instead

                print $wfh $_ if /\bterminal length\b/i .. /^\n*$/ ;
            }
        }
        else {
            print "auth failed";
        }
    }
}
else {
    print "not pinging";
}
[download]

update: added the default var after the print to the file handle, per Re^3: download as txt and pdf.