Absolutely YES (you can do high speed flash work with any camera).
First, speedlights are fast (regular camera flash is called Speedlights). Up close, at low power, speedlights are incredibly fast. 1/64 power level (maybe 3 feet at ISO 400) has a flash duration of maybe 1/30,000 second. This is used for high speed photography, milk drop spashes, bursting water balloons, hummingbird wings, etc. For routine bounce flash at 1/2 power, still about 1/1000 second duration. Speedlights stop motion and action.
And second, HSS is anything but fast. Speedlight is the first way to go, and HSS is the absolute last choice.
HSS is High Speed Sync, but it is NOT high speed flash. HSS becomes a continuous light (for the duration of the shutter travel), and continuous light (like sunlight) cannot stop motion at all. It merely allows any sync speed, removes all sync requirements (called High Speed Sync), so the camera shutter can be fast... But a fast camera shutter simply decimates continuous light. A 1/1000 second shutter sees only 1/1000 of the light that a 1 second shutter would see. Like sunlight, we can compensate by opening the aperture. This is what allows using f/2.8 in bright sunshine, with HSS flash too. But the HSS concept is ANYTHING BUT FAST. HSS does allow fast shutter, but a speedlight is faster than camera shutters can be.
In order for HSS to be continuous, it has to run at about 20% power level. This of course has less range than speedlights which can run at full power level.
Auto FP and HSS - What is it? is more about HSS.
The Nikon Commander (which the D5500 does not have) can trigger remote HSS flashes. The camera internal flash cannot do HSS, but it can be used as Commander to trigger HSS flashes (if the internal flash is disabled from contributing to the scene lighting).