"Linear" imposes a set of restrictions. "Non-linear" simply means there are no restrictions.
Many non-linear control schemes can be faster than linear ones. Linear control schemes are restricted to "smoothly" transitioning. Non-linear control can be implemented by suddenly slamming a digital value, for example.
A good example of a fast-responding non-linear control scheme is a typical thermostat. When the room is below the set point, the heater is turned on fully. When above the set point, the heater is turned off fully.
This is obviously the fastest way to get to the desired temperature. Such a system may be less "good" than a linear controller in other metrics, like overshoot, but that's not what you asked about.
The reason linear control schemes are sometimes chosen is not for their speed or other control property, but because they can be mathematically analyzed more easily. There has been much theory and techniques developed around linear systems that don't apply to non-linear systems. Consider Laplace transforms and S-parameter analysis, for example.