The answer to your question (assuming there is one) needs to be approached indirectly.
A key question to ask might be,
"Does the Bible indicate that animals possess a soul or spirit?"
There are some indications in Scripture they do.
Genesis 1:30 - "'And to all the animals of the earth, and to every bird of the air, and to all the creatures that move on the ground – everything that has the breath of life in it – I give every green plant for food.' It was so."
Leviticus 24:18 - "One who beats an animal to death must make restitution for it, life for life."
Eccelsiastes 3:19 - "For the fate of humans and the fate of animals are the same:
As one dies, so dies the other; both have the same breath. There is no advantage for humans over animals, for both are fleeting."
Jonah 4:11 - "'Should I not have compassion on Nineveh, the great city in which there are more than 120,000 persons who do not know the difference between their right and left hand, as well as many animals?'"
It is obvious that human beings and animals have at least a few things in common:
I. The same Creator created them both.
II. The same breath of God which animates creatures created in God's image is the same breath which animates the animals.
III. While animals do not have the same value in God's estimation as the human beings whom He loves, God still has compassion on animals (see Jonah 4:11, above). Jesus said,
"'Look at the birds of the air, that they do not sow, nor reap nor gather into barns, and yet your heavenly Father feeds them. Are you not worth much more than they?'" (Matthew 6:26 NASB Updated).
IV. After the fall of our first parents, the animal kingdom felt the repercussions of that fall in physical death. For that matter, Paul tells us, all of creation
"was subjected to futility, not willingly, but because of Him who subject it, in hope that the creation itself also will be set free from its slavery to corruption into the freedom of the glory of the children of God" (Romans 8:20-21 NASB Updated).
Neither the death of human beings nor the death of animals was part of God's plan from beginning. If it weren't for sin, humans and animals would have lived forever. Because of sin,
"we know that the whole creation groans and suffers the pains of childbirth together until now" (ibid., v.22).
The good news, however, is that a day of freedom is coming, and nothing can stop it. When it finally comes, we human beings will experience the redemption of our bodies (ibid., v.23). Animals, too, will experience liberation from death. As Narnian above has already observed, Isaiah predicted a day when animals would no longer be either prey or predator, but would live at peace with one another and with the human race as well (see Isaiah 11:6-8 NASB, and Isaiah 65:25 NASB).
Years ago, I heard a famous preacher admit openly that on more than one occasion he wondered if there was going to be golf in heaven. Somehow he just couldn't picture life in heaven without his favorite game on earth! The answer which came to him was simply this (and I paraphrase): Why do you think that just because you'll be in heaven, God would of necessity put the kibosh on your having a good time doing something you enjoyed intensely while on earth? Why wouldn't God want you to enjoy yourself in that way?
Where did we get the notion that all we'll do in heaven is "attend church and play our harps of gold"? Certainly not from the Bible. What did Paul say in 1 Corinthians chapter 2?
"But as it is written, Eye hath not seen, nor ear heard, neither have entered into the heart of man, the things which God hath prepared for them that love him. But God hath revealed them unto us by his Spirit: for the Spirit searcheth all things, yea, the deep things of God" (vv.9-10 KJV).
If with the help of the Holy Spirit we look deep within our spirits and exercise our sanctified imaginations, we'll begin to realize God has no intention of depriving us of anything in heaven. Will there be times of praise in heaven? Yes. In addition to a few harps in heaven's orchestra, there will also be more than a few trumpets, guitars, violins, doublebasses, flutes, clarinets, tympani, cymbals, organs, and maybe even a didgeridoo! God is a God of variety and creativity. There will probably be instruments in heaven we have never even heard of, let alone know how to play! But we'll learn.
ALL THIS TO SAY: If a pet animal was very dear to you on earth, what makes you think you will never again see little Rover or little Miss Kitty? Let's not put limitations on God just because there isn't a verse in the Bible somewhere which says explicitly and unambiguously, "All pets go to heaven."
Now the argument could be made that on this side of heaven, what we deem to be so special and impossible to live without, God will replace in heaven with something much, much better, inconceivably better, even. Just because the argument can be made, however, does not make something either true or false.
I'm sure God is not the least bit offended by the sentiment expressed by Hank Williams, Jr., in one of his songs, though Williams's theology does need a little fine-tuning:
"If heaven ain't a lot like Dixie, then I don't wanna go."
In conclusion, can we say with biblical certainty that animals, like humans, have an immortal soul or spirit? I don't know. According to Dr. J.P. Moreland, a scientist, philosopher, and theologian, they may not.
"[Animals] are not simply machines. They have consciousness and points of view. But the animal soul is much simpler than the human soul. For example, the human soul is capable of free moral action, but I think the animal soul is determined. Also, Augustine said animals have thoughts, but they don't think about their thinking. And while we have beliefs about our beliefs, animals don't. . . . The human soul is vastly more complicated because it's made in the image of God. So we have self-reflection and self-thinking. And while the human soul survives the death of its body, I don't think the animal soul outlives its body. I could be wrong, but I think the animal soul ceases to exist at death" (see Lee Strobel, The Case for a Creator; Grand Rapids: Zondervan, 2004, pp.262, 263).
Personally, I think Dr. Moreland is wrong. From my perspective, at least at present, there is no particular reason why the soul of a human being can exist forever, but the soul of an animal cannot. Soul is soul, and if a human being's identity derives from something invisible which is inherently more than just the product of brain chemistry, why must an animal's identity stop at the brain, no matter the size?