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CTR prediction has been widely used in the real world. Many methods model feature interaction to improve their performance. However, most methods only learn a fixed representation for each feature without considering the varying importance of each feature under different contexts, resulting in inferior performance. Recently, several methods tried to learn vector-level weights for feature representations to address the fixed representation issue. However, they only produce linear transformations to refine the fixed feature representations, which are still not flexible enough to capture the varying importance of each feature under different contexts. In this paper, we propose a novel module named Feature Refinement Network (FRNet), which learns context-aware feature representations at bit-level for each feature in different contexts. FRNet consists of two key components: 1) Information Extraction Unit (IEU), which captures contextual information and cross-feature relationships to guide context-aware feature refinement; and 2) Complementary Selection Gate (CSGate), which adaptively integrates the original and complementary feature representations learned in IEU with bit-level weights. Notably, FRNet is orthogonal to existing CTR methods and thus can be applied in many existing methods to boost their performance. Comprehensive experiments are conducted to verify the effectiveness, efficiency, and compatibility of FRNet.