Abstract In this paper, we locally classify the surfaces immersed into the non-flat (Riemannian or Lorentzian) 3-space forms satisfying the condition $$\Box \vec {\textbf{H}}=\lambda \vec {\textbf{H}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>□</mml:mo> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:mrow> </mml:math> for a real number $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> , where $$\vec {\textbf{H}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:math> is the mean curvature vector field and $$\Box $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>□</mml:mo> </mml:math> denotes the Cheng–Yau operator of the surface. We obtain the classification result by proving, at a first step, that the mean curvature function must be constant and, in a second step, we complete the classification.