While the question of whether animals continue to exist in some form after death has yet to be answered scientifically one way or the other, I believe the answer is yes. I believe animals have a soul that continues on after they have passed.
A well known conception of animal afterlife is described in The Rainbow Bridge, a poem by an anonymous writer. The poem depicts a beautiful meadow where pets who have passed away play until their people come to join them, and together they cross over the Rainbow Bridge. I think it's safe to say that pet parents would hope for such a place for their departed pets, and add in all the favorite foods and toys that their pets enjoyed in life.
Anecdotal evidence of an afterlife for animals does exist. Some of this comes from accounts by people who have had a near death experience (NDE).
An article by Scott Jansson shares the story of Alma, an elderly women who explained she was not afraid of dying, since she had briefly glimpsed the afterlife when she was younger. Alma was a survivor of a fire. She remembers leaving her body and looking down to see the EMTs working on her. She then followed a beautiful light and music to a yard from her childhood. Her dog Sadie was there, and greeted her with joy. Alma's mother joined them as well, and told Alma she had to go back. The presence of Sadie in this NDE suggests that animals are part of the afterlife.
A similar phenomenon has occurred in which people who are near death have dreams or waking visions of the afterlife.
Dr. Christopher Kerr wrote about people experiencing End of Life Dreams and Visions, or ELDVs in his book Death is but a Dream: Finding Hope and Meaning at the End of Life. According to Dr. Kerr, it is especially common in children who are near death to have these NLDVs include their pets who are no longer living. One example he gives is of 13 year old Jessica, who at the end of her life had dreams about her previous pet, a black Lab named Shadow.
Many people have recounted stories of visitations from their pets who have passed away. These visitations can come in many forms: a sense that the animal is near, a physical feeling of them walking by or leaning against their person, an image. Synchronicities can also symbolize that a pet is reaching out to their person. You might come across another pet with the same name as yours or see another animal doing something that was uniquely characteristic of your pet. Yes, these could simply be coincidences, or they could be a hello from your departed pet.
Dr. Adrian Klein, an afterlife researcher, wrote a case report about evidence of his beloved mini Schnauzer, Sally, existing after her death. Sally passed away at the age of 17 in October 2013. Dr. Klein received messages from Sally through an animal communicator and medium, who was able to describe Sally's transition to heaven, as well as some of the medical procedures she had undergone prior to her passing.
Sally also appeared to Dr. Klein through video imaging known as the Schreiber Technique. This technique basically entails taking video of random spots such as a puddle, or curtain, or the corner of a room. When going through the video frame by frame, sometimes images of those in spirit can be seen. Dr. Klein's case report includes multiple images of Sally, some alone and some with other departed loved ones.
Very close to the idea that animals have an afterlife is the experience some people have had with an animal reincarnating.
One of these is the story of Rina, a Golden Retriever/Labrador mix. Rina's person, Judy Fridono, shares her amazing experience of Rina not only sending her signs shortly after her death, but eventually returning to her as another puppy. Judy's blog chronicles the whole story, and offers hope and comfort that our pets do have souls and do continue to exist beyond death.
Following this thread further, I do wonder whether this means non-domesticated animals also have an afterlife. Is it just the animals who come into the lives of humans? What about all the animals who are part of our food industry? What about insects? Clearly this is one of those topics that leads to more mysteries yet to be solved. Being open to wonder about it though, can alter one's perspective on life and death.
There is nothing equal to having our pets here in the living with us. To know that their soul continues on does not take away the excruciating grief of having to let them go. Perhaps it brings some comfort. To imagine them healthy, vibrant, and happy, romping about near the Rainbow Bridge, or maybe sticking by our side, still giving companionship and love everlasting.