Depends...
What kind of LED, particularly how much power?
Not a white LED I hope/assume, because then all bets are off - most white LEDs rely on phosphor coatings that are way slower than what you need.
The higher the power, the more critical the driver circuitry would become.
That's slooooooow for an Arduino and semiconductors. Don't worry about it. Especially if you can get away with using some kind of linear current limiter circuit (e.g. a simple resistor).
How about a PCA9685. That's something like 1.4kHz@ 12 bits, hence something like 0.71ms / 4095 = ca. 173ns shortest pulse width.