This won't make it any faster, but you can avoid having to use the bigint package by reducing the factorial mod $prime every time you add a factor:

my $r = 1;
my $k = 2;
while ($k < $prime) {
  $r = ($r * $k) % $prime;
  $k++;
}
if ($r + 1 % $prime == 0) {
  print "prime\n";
} else {
  print "not prime\n";
}
[download]

Then $r never gets bigger than $prime.