It is, at this point in time, unknown as to whether Archimedes had any students or ran a school-(similar to Aristotle's Lyceum or Plato's Academy). If I was to provide an educated guess-(based upon the little that I know regarding Archimedes' biography), I would say......probably not. By today's standards, Archimedes would have been a professional Scientist and Inventor similar to someone, such as Thomas Edison. Due to his plethora of inventions and mathematically oriented writings, I am not sure Archimedes would have had the time to teach! He may have had his own version of Teaching or Research Assistants-(similar to Aristotle's Lyceum), though, based on our current evidence, it seems to have been unlikely.
However, I don't think it's an impossibility that Archimedes may have had a school or a group of students in Siracusa, Sicily or that perhaps Archimedes may have had the ancient equivalent of a Visiting Professorship or Teaching Fellowship at The Alexandrian University in Egypt. Many of the major Greek Scientific and Philosophical Thinkers dating back to Thales-(The Father of Western Science and Philosophy), were also Teachers and Educators. Democritus, Pythagoras, Zeno, Heraclitus, Socrates, Hippocrates, Plato, Aristotle, Epicurus and Hypatia, were all professional Teachers and Educators, while also pursuing scientific and/or philosophical writings and research. So it is not necessarily an impossibility for Archimedes to have been part of the centuries long Ancient Greek intellectual tradition.
However, at this point in time, there is just no primary evidence which supports such a theoretical or circumstantial claim and therefore, it continues to remain unknown.