Yes. The basic technique is to create a new package which inherits from the old one but adds some new methods, then rebless the object into that new package. Here's a quick example:

use 5.010;
use Math::BigInt;

# quite big!!
my $seven = Math::BigInt->new('7');

# $seven cannot "speak", so this warns
eval { $seven->speak } or warn $@;

{
    package Math::BigInt::Speaker;
    our @ISA = qw(Math::BigInt);
    use Scalar::Util qw(blessed);
    use Carp qw(confess);
    sub upgrade {
        my ($class, $instance) = @_;
        confess "can only upgrade Math::BigInt objects"
            unless blessed $instance && $instance->isa('Math::BigInt')
+;
        bless $instance => $class;
    }
    sub speak {
        CORE::say($_[0]);
    }
}

# Swaps $seven into our new class.
Math::BigInt::Speaker->upgrade($seven);

# $seven can now "speak"
eval { $seven->speak } or warn $@;
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If you use Moose you can do this in a much nicer and more organised way. (Though it's basically the same thing happening under the hood.) You'd just create a new anonymous role, and apply that role to the object.

use 5.010;
use Math::BigInt;
use Moose ();

# quite big!!
my $seven = Math::BigInt->new('7');

# $seven cannot "speak", so this warns
eval { $seven->speak } or warn $@;

my $speaker = Moose::Meta::Role->create_anon_role(
    methods => {
        speak => sub { CORE::say($_[0]) },
    },
);

Moose::Util::apply_all_roles($seven, $speaker);

# $seven can now "speak"
eval { $seven->speak } or warn $@;
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