Can I be reincarnated as an animal as some religions say?

No, there is a progressive law, and though our Spirit has been worked out in lower realms, we do not return to them.
«The soul sleeps in the mineral, dreams in the vegetable, moves in the animal and awakens in man»., decía Léon Denis.
The Spirits' Book, in questions 607 and 609, explains that the Spirit comes from the intelligent principle which is elaborated in these lower kingdoms, and that in the first incarnations as Spirit it retains vestiges of that animality. The intelligent principle to become spirit passes through other species, through other forms and lives which we are still far from knowing, because there are missing links, missing pieces in this evolutionary puzzle, which do not exist today because they have given way to others by natural selection, and because this evolution does not occur only on this planet.
Taking into account only our planet and what Science says today, there is an evident biological evolution, a natural selection, it is what is known today as the synthetic theory that groups the theories of Darwin and Wallace with those of Mendel. And what Spiritism says is that the intelligent principle is elaborated in these and other different species of this and other worlds, which are also inhabited in different evolutionary processes, such as those that our planet has undergone since the prokaryotic cell. There is therefore a physical evolution subject to physical laws, and a spiritual evolution which uses this physical evolution to evolve the intelligent principle, finitely the Spirit, through reincarnation.
Many religions have and have had reincarnation as one of their fundamental principles, although under erroneous aspects such as metempsychosis (which would be the possibility of incarnating in animals). Other religions only had reincarnation in their beginnings, being eradicated later, as is the case of Christianity.