Given a boolean matrix A of size n, A^p can be computed in space O(log n log p)
No.  The intermediate matrices $A^{2^t}$ don't need to be stored anywhere.  When you need a value, you just recompute it.  This might sound very inefficient -- but remember, we're only counting the space complexity here, not the time complexity, so it's fine to take a very long time to recompute the same value many times, if that saves you some space.