  We consider the noncommutative extension of the BF theory in two spacetime
dimensions. We show that the introduction of the noncommutative parameter
\theta_{\mu\nu}, already at first order in the analytical sector, induces
infinitely many terms in the quantum extension of the model. This clashes with
the commonly accepted rules of QFT, and we believe that this problem is not
peculiar to this particular model, but it might concern the noncommutative
extension of any ordinary quantum field theory obtained via the Moyal
prescription. A detailed study of noncommutative anomalies is also presented.
