  We consider Riesz transforms of any order associated to an
Ornstein--Uhlenbeck operator $\mathcal L$, with covariance $Q$ given by a real,
symmetric and positive definite matrix, and with drift $B$ given by a real
matrix whose eigenvalues have negative real parts. In this general Gaussian
context, we prove that a Riesz transform is of weak type $(1,1)$ with respect
to the invariant measure if and only if its order is at most $2$.
