[A2A] I have never personally seen an IAS pursuing higher education from IIT, but it should be possible based on the following observations. IITs usually host a few students in each department who come from military or DRDO or other government agencies or even private agencies. IAS officers are sponsored by the government to pursue higher studies abroad, so they should be happy if someone wants to pursue their higher studies in IITs as it would cost less. IITs would be happy to have an IAS officer doing M.Tech./Ph.D. as it adds to their "resume."
Kiran Bedi (not an IAS, but IPS) got a Ph.D. from IIT Delhi in 1994 while still in service. Kiran Bedi