I am not sure why the code terminates, however I do have a comment about your code.

You open a handle to the log file each time you process a host from your @hostnames array. Perhaps you could look at opening the file handle once before processing the entire array. Also you may want to only sleep after the foreach...

Something like

use strict;

use Time::Local;
use Net::Ping;

my ($time);
my @hostnames    = ("10.0.0.1", "10.0.0.2");
my $p        = Net::Ping->new("icmp");
my $hostname;

 print "\nWriting data to ping.log\n";
 print "Press CTRL-C to exit\n";

while(1)
{
    open (LogFile, ">>ping.log") || die "Sorry, cannot open logfile\n"
+;

    foreach my $hostname (@hostnames)
    {
        $time = localtime;
        print (LogFile "$time - ");
        print (LogFile "$hostname is ");
        print (LogFile "NOT ") unless $p->ping($hostname, 2);
        print (LogFile "reachable.\n");
     }
     close Logfile;
     sleep(5);    
}
[download]

These changes will not affect you unless you @hostnames array grows to more than 1 host though.