Yes, an atom consists of even smaller particles. At the center of an atom, you have a nucleus composed of protons and neutrons. Protons have a positive charge, while neutrons have no charge. Around the nucleus, there is an electron cloud where electrons, which carry a negative charge, are found.
While atoms can be further broken down into smaller particles, such as protons, neutrons, and electrons, the typical chemistry lab doesn't have the capability to break down atoms into their constituent parts. The study of breaking down atoms into their fundamental particles falls under the field of particle physics.
However, in certain specialized laboratory settings, such as high-energy physics experiments or particle accelerators, scientists can create conditions where they can study the behavior of subatomic particles and even create new particles by colliding atoms together at high speeds. But this is done in highly controlled and specialized environments that go beyond the scope of a typical chemistry lab.