I'm backing you up on this one. Although, it is generally considered unclear exactly what happens to animals in general when they die, as far as the bible is concerned, I am pretty sure there aren't any verses that contradict the "animals have spirits" ones that explicitly state that animals don't have spirits.
I've spent 20 years studying the Bible, both in Hebrew and Greek. No, I don't speak those languages, but I have always looked up words that we take for granted in the Hebrew and Greek to see how they thought of the words.
Both the word Ruach (breath, wind, spirit) and nephesh (a soul, living being, creature, etc) are applied to both man and animals. Not to plants ever.
Since they are applied to both man and animals, then I figure they can not be what uniquely made man in God's image. How can they be if the animals posses it too?
The only verse that applies something unique to man, and only man is Genesis 3:22 where yada (to know) is applied in relation to man has become (hayah) like unto God.
Which also is why I argue against a young earth creation view, as hayah (here everyone translates it has become) but in verse 2 of chapter one they translate it "was". So the earth "became" desolate and waste..." But that's another subject and I digress.
But yada in relation to becoming like unto God is the only unique occurrence of any usage in the Bible of relating man to God. So no, I don't believe we have a soul, or anything special apart from the animals except in our ability to reason or understand. To make a conscious choice in how we live our lives and to believe or not to believe. Hence Jesus called "teacher" by the disciples, or the Word. Logos denotes Knowledge, wisdom, thought. An idea.
https://www.biblestudytools.com/lexicons/greek/nas/logos.html
I mean a computer is technically smarter than me, but it can't understand. It can analyze, but it can't understand or even know as we use the term. Computers and animals can not yada.