Try this. Notably faster on my machine that is horribly slower than yours.

package Organism;

use strict;
# use warnings;

sub count {
   return scalar keys %{ shift->{Cells} };
}

# Input a list of [ x, y ] coords
sub insert_cells {
   my $cells = shift->{Cells};
   for my $r (@_) { $cells->{ pack 'i2', @{$r} } = undef }
}

# Return sorted list of cells in the Organism.
# Used for verification and testing the state of the organism.
sub get_live_cells {
   sort { $a->[0] <=> $b->[0] || $a->[1] <=> $b->[1] }
      map { [ unpack 'i2', $_ ] }
      keys %{ shift->{Cells} };
}

sub tick {
   my $self  = shift;
   my $cells = $self->{Cells};
   my ( $k1, $k2, $k3, $k4, $k5, $k6, $k7, $k8,
        $x0, $x1, $x2, $y0, $y1, $y2, %new_cells, %dead_cells );

   for my $c (keys %{ $cells }) {
      # Get the (up to 8) dead cells surrounding the cell
      ( $x0, $y0 ) = unpack 'i2', $c;
      ( $x1, $x2, $y1, $y2 ) = ( $x0 - 1, $x0 + 1, $y0 - 1, $y0 + 1 );
      $dead_cells{$_}++ for my @zcells = (
         ($k1 = pack 'i2', $x1, $y1) x !(exists $cells->{$k1}),
         ($k2 = pack 'i2', $x1, $y0) x !(exists $cells->{$k2}),
         ($k3 = pack 'i2', $x1, $y2) x !(exists $cells->{$k3}),
         ($k4 = pack 'i2', $x0, $y1) x !(exists $cells->{$k4}),
         ($k5 = pack 'i2', $x0, $y2) x !(exists $cells->{$k5}),
         ($k6 = pack 'i2', $x2, $y1) x !(exists $cells->{$k6}),
         ($k7 = pack 'i2', $x2, $y0) x !(exists $cells->{$k7}),
         ($k8 = pack 'i2', $x2, $y2) x !(exists $cells->{$k8}) );

      # Check the live cell
      # Note: next line equivalent to nlive == 2 || nlive == 3
      @zcells == 5 || @zcells == 6 and $new_cells{$c} = undef;
   }
   $dead_cells{$_} == 3 and $new_cells{$_} = undef for keys %dead_cell
+s;
   $self->{Cells} = \%new_cells;
}

sub new {
   my $class = shift;
   my %init_self = ( Cells => {} );
   bless \%init_self, $class;
}

1;
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