This is difficult to answer, we simply cannot perceive what animals feel or understand. However, from our limited understanding, and what we define as spirituality, the most obvious answer is that they do not.
Unfortunately there is no way we as humans can be sure. Animals very well may be close to humans. In that sense they could have a sense of spirit as part of their lives. Since animals cannot speak there is not a true way to depict for sure if they do.
