Can I make physics engine speed dependent on frame rate?

By default physics engine in Unity is completely independent on the frame-rate - it depends on time delta between updates, so the game plays at the same speed regardless of the frame rate.

My game is frame-rate dependent - it is meant to played at exactly 60FPS (whether that is right or wrong thing to do is out of scope of this question) and should slow down in case of an occasional drop of frame rate. The physics engine does not obey, so when everything slows down, objects which are moved by the physics engine appear to speed up.

Is it possible to force the physics engine to always assume that a fixed amount of time has passed between frames?

About the only way I can see to do this is to change the timescale every frame based on the frame rate. This will probably give you other problems.

Yes it is possible. You want to use the Time Manager

You might want to do some of your other code in FixedUpdate() too.