Exercising an option the same day of buying it (regardless of time value)

Can I exercise an option the minute after I buy it?

I am quite experienced with options and know my intrinsic/time value components, however, if there is an opportunity I see where options are mispriced and underlying is illiquid, do I have to wait for T+1 settlement date for options? or is T+1 settlement only refers to time between assignment/exercise of option to delivery of underlying?

Any help is greatly appreciated!!