Can I be a student and a SERE specialist?

If you're asking if you can enlist to be a SERE Instructor, unless your last name is Norris, and your first name is Chuck, that is a no.
All is not lost though, there are some Military Occupational Specialties (MOS) that have a greater chance of attending the SERE course, and your Recruiter can identify those.
To be clear, few MOSs have the need to attend the course, and far fewer become SERE instructors.

PFC (Join to see) Read through the following material very carefully. Then read through it again and make a list of questions to ask a Recruiter. Joining the active duty military is a serious commitment, and they are not going to send you to SERE training unless you qualify, and are committed. Once you complete all of your initial training, you could attend classes on your own time as time permits.

If you want to go to school, and pursue a civilian career, you could join the Reserves or Guard, but being selected for a SERE slot might be difficult. Again read all of this material and then talk to a Recruiter. You will not be considered for expensive training unless you are considered a valuable dedicated member of your unit. School, and a civilian career would probably have to be on hold until you finish a year or more of training.

Are you in high school or college, are you working, and if so, what is your job?

Read the material, talk to a recruiter, and provide us with an update.


https://www.thebalancecareers.com/usaf-sere-career-profile-2356491

https://www.fairchild.af.mil/News/Article-Display/Article/239639/airmen-graduate-from-first-guard-and-reserve-sere-course/

https://bootcampmilitaryfitnessinstitute.com/elite-special-forces/us-elite-special-forces/afsoc-us-air-force-special-operations-command/us-air-force-sere-specialist-selection-training/

When I was serving active duty they said I could attend "outside schooling" if I wanted to, if time permitted.......yep, time never permitted for 7 1/2 years.
Being active duty, you will most likely NEVER have time for outside schooling. Field training, classroom training, duty requirements, etc. will always prevent that. You should pretty much decide if you want school OR Active Duty Military...from what I have seen, you can't have both at the same time....prioritize what you want to do.
Do one, then the other, but don't plan on doing both at the same time. It's practically impossible to do both.
(based on experiences in the Army branch)