Domain Adaptive Object Detection (DAOD) aims to transfer detectors from a labeled source domain to an unlabeled target domain. Existing DAOD methods employ multi-granularity feature alignment to learn domain-invariant representations. However, the local connectivity of their CNN-based backbone and detection head restricts alignment to local regions, failing to extract global domain-invariant features. Although transformer-based DAOD methods capture global dependencies via attention mechanisms, their quadratic computational cost hinders practical deployment. To solve this, we propose DA-Mamba, a hybrid CNN-State Space Models (SSMs) architecture that combines the efficiency of CNNs with the linear-time long-range modeling capability of State Space Models (SSMs) to capture both global and local domain-invariant features. Specifically, we introduce two novel modules: Image-Aware SSM (IA-SSM) and Object-Aware SSM (OA-SSM). IA-SSM is integrated into the backbone to enhance global domain awareness, enabling image-level global and local alignment. OA-SSM is inserted into the detection head to model spatial and semantic dependencies among objects, enhancing instance-level alignment. Comprehensive experiments demonstrate that the proposed method can efficiently improve the cross-domain performance of the object detector.