In this paper we prove the existence of solutions to the periodic BVP on [0,p] \((p>0)\) for the second order vector differential equation of Liénard type in \({\mathbb{R}}^ m\) \(x''(t)+(d/dt)\phi (x(t))+Ag(t,x(t))=h(t)\) (t\(\in [0,p])\), with \(\phi:=\text{grad} F\), \(F\in C^ 2({\mathbb{R}}^ m,{\mathbb{R}})\), A an \(m\times m\) constant matrix, g: [0,p]\(\times {\mathbb{R}}^ m\to {\mathbb{R}}^ m\) satisfying the Carathéodory conditions and h: [0,p]\(\to {\mathbb{R}}^ m\) (Lebesgue) integrable. The main assumption is a sharp non-resonance condition on the nonlinear field g, with respect to the first two eigenvalues 0 and \(\omega^ 2=(2\pi /p)^ 2\) of the linear operator \((-d^ 2/dt^ 2)\) subjected to the periodic boundary conditions on [0,p]. Thus, a theorem by R. Reissig [Abh. Math. Semin. Univ. Hamb. 44, 45-51 (1975; Zbl 0323.34033)], concerning the scalar Liénard equation, is extended to the vector case and improved. Some kind of non-uniform non-resonance condition [see J. Mawhin and J. R. Ward, Arch. Math. 41, 337-351 (1983; Zbl 0537.34037)] is also considered. All the results also apply to the retarded Liénard equation \(x''(t)+(d/dt)\phi (x(t))+Ag(t,x(t- \tau))=h(t),\) with \(\tau\in [0,p[\).