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Deep learning has attracted increasing attention across a number of disciplines in recent years. In the field of remote sensing, ship detection based on deep learning for synthetic aperture radar (SAR) imagery is replacing traditional methods as a mainstream research method. The multiple scales of ship objects make the detection of ship targets a challenging task in SAR images. This paper proposes a new methodology for better detection of multi-scale ship objects in SAR images, which is based on YOLOv5 with a small model size (YOLOv5s), namely the multi-scale ship detection network (MSSDNet). We construct two modules in MSSDNet: the CSPMRes2 (Cross Stage Partial network with Modified Res2Net) module for improving feature representation capability and the FC-FPN (Feature Pyramid Network with Fusion Coefficients) module for fusing feature maps adaptively. Firstly, the CSPMRes2 module introduces modified Res2Net (MRes2) with a coordinate attention module (CAM) for multi-scale features extraction in scale dimension, then the CSPMRes2 module will be used as a basic module in the depth dimension of the MSSDNet backbone. Thus, our backbone of MSSDNet has the capabilities of features extraction in both depth and scale dimensions. In the FC-FPN module, we set a learnable fusion coefficient for each feature map participating in fusion, which helps the FC-FPN module choose the best features to fuse for multi-scale objects detection tasks. After the feature fusion, we pass the output through the CSPMRes2 module for better feature representation. The performance evaluation for this study is conducted using an RTX2080Ti GPU, and two different datasets: SSDD and SARShip are used. These experiments on SSDD and SARShip datasets confirm that MSSDNet leads to superior multi-scale ship detection compared with the state-of-the-art methods. Moreover, in comparisons of network model size and inference time, our MSSDNet also has huge advantages with related methods.