I believed that matrix multiplication and squaring to be significantly
faster over GF(p) instead over ZZ.

Numerical evidence supports the opposite, is it a bug or reality?

Session:

def randmat(N,K):  return Matrix(K,N,N,[K.random_element() for _ in range(N^2)])
N=200
Kq=GF(next_prime(2^200))
M1=randmat(N,Kq)
%time MsZZ=M1.change_ring(ZZ)^2 #Wall time: 188 ms
%time Msqu=M1^2 #Wall time: 7.75 s

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