Yes. With most representatives of the breed, however, this is not particularly pronounced and can often be neglected. Nevertheless, a Chihuahua is and remains a dog and certain instincts can never be completely bred away.
Accordingly, there are also specimens of the smallest breed in the world that run after and hunt rodents, birds, rabbits, or even cats. The pursued animal can be injured or killed.
If there are other animals in the same household, caution and good observation skills are required.