Were there any successful flying machines that did not rely on Bernoulli's Principle?
How about all lighter-than-air vehicles:


Hot air balloons
Hydrogen- or helium-filled balloons
Airships


If you want to restrict the choice to heavier-than-air vehicles, rockets should still qualify.

You may also include VTOL aircraft which do not rely on their wings during hover, their internal turbo machinery would still use Bernoulli's principle, however, so for me they do not qualify.