<p>In the analysis of problems related to size effects, the flexibility and versatility demonstrated by polygonal finite elements are particularly essential. This paper proposes a new arbitrary polygonal couple stress element method for both linear and geometrically nonlinear analyses. Based on the modified couple stress theory, 23 fundamental analytical stress function solutions are derived and incorporated into a hybrid stress function (HSF) framework. An arbitrary <i>n</i>-sided polygonal couple stress element HSF-PCS is developed in this framework. Independent drilling degrees of freedom and an innovative penalty function are introduced to satisfy higher-order continuity requirements. The penalty term ensures that the independently assumed rotational field and the physical rotational field are equal in a piecewise averaged sense at the element's centroid, thereby avoiding the challenges of <i>C</i>¹ displacement interpolation within polygonal elements. Then, the original element formulation is extended to geometric nonlinearity through the incorporation of an incremental corotational method and updated analytical solutions of stress functions. Numerical examples show that the proposed element demonstrates superior accuracy and flexibility compared to traditional quadrilateral elements, particularly for complex geometries and stress concentration problems.</p>