The problem is you are using a package-scoped variable ($_name) to store instance data instead of storing it in the object itself. Your methods should be closer to

  sub myNameIs{
    my ($this,$name) = @_;
    die("Don't tell me that I don't have a name.") unless(defined($nam
+e));
    
    $this->{_name} = $name;
  }

  sub whoAmI{
    my $this = shift;
    return($this->{_name});
  }
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See perltoot for a tutorial of Perl OOP.