Can an airplane engine fire be extinguished without shutting down the engine? Airplane engines catch fire from time to time (and are falsely reported as being on fire even more often).  In a modern airliner, the "fire handle" shuts down the engine, closes the fuel pipe, and discharges the fire extinguisher.

Occasionally it's desirable to leave the burning engine running because you need the added thrust. For example during WWII B-29 engines had an unfortunate habit of catching fire on takeoff. Standard procedure was to leave the burning engine running until you got enough altitude to safely discard the bomb load and/or bail out.

Is it possible to activate the fire extinguisher without shutting down the engine?
In most modern jets (e.g. Boeing 757/767) when you pull the fire handle the engine shuts down (fuel, hydraulic, etc., are cutoff). But pulling the fire handle only "arms" the extinguishing system.  You have to rotate the handle (left or right) to discharge a bottle. (there are other bottle configurations based on the airplane type, but this is the general scheme)

Usually, after pulling the fire handle (before rotating it left or right) you wait momentarily to determine if the overheat or fire warning lights have extinguished.  If not, you rotate the handle to discharge the extinguisher.

So, if you are asking about modern jets, the answer would be no, you can't discharge the extinguishing agent without shutting down the engine first.

I am not aware of a modern design that would allow the engine to continue operating after activating the fire extinguishing system.