No, it says they have a nephesh...
Animals have a soul.
Man's soul was created in God's image.
Not so, for animals.
There is a difference.
In Genesis One describes all the souls were created.
(bara = created out from nothing)
In Genesis Two, bodies were then provided for those souls.
(jatsar = bodies molded and formed from the elements of the earth)
Yes... animal souls, too, were provided bodies 'molded and formed' -jatsar, out from the earth!
Not just Adam's body!
Now the Lord God had formed out of the ground all the wild animals and all the birds in the sky.
He brought them to the man to see what he would name them; and whatever the man called each
living creature, that was its name. Genesis 2:19
One could assume that the sea creatures were made of the elements taken from lakes and oceans.
God has a pattern He follows.