Perhaps I might rephrase the question to facilitate clarity of my answer:

Through cognitive processes, can we acquire information faster than can be conveyed at the maximum known mass carrier velocity, light speed?

The answer is a provable - Yes.

To want to define what is meant by "speed of mind" is to miss the spirit of the original question. The mind is often related to reasoning and perception and as with all processes, requires time to execute. This is colloquially referred to as "speed", even though no distance is travelled by the mind. That is, the time it takes for one to come to an understanding, or whatever other result is expected given the circumstances, is "speed of mind".

Relating this to light speed is simply a comparative measure with time elapsed at the fastest mass velocity, to complete the same operation.

Many common expressions use the term "speed" and "how fast" something happens without any travel distance involved.

Some minds though are slower than others. Snails for example.