Quarkonium production at low transverse momentum in hadron collisions can be used to extract Transverse-Momentum-Dependent (TMD) gluon distribution functions, if TMD factorization holds there. We show that TMD factorization for the case of P-wave quarkonium with J(PC) = 0(++), 2(++) holds at one-loop level, but is violated beyond one-loop level. TMD factorization for other P-wave quarkonium is also violated already at one-loop level. (C) 2014 The Authors. Published by Elsevier B.V.