The automatic segmentation of COVID-19 pneumonia from a computerized tomography (CT) scan has become a major interest for scholars in developing a powerful diagnostic framework in the Internet of Medical Things (IoMT). Federated deep learning (FDL) is considered a promising approach for efficient and cooperative training from multi-institutional image data. However, the nonindependent and identically distributed (Non-IID) data from health care remain a remarkable challenge, limiting the applicability of FDL in the real world. The variability in features incurred by different scanning protocols, scanners, or acquisition parameters produces the learning drift phenomena during the training, which impairs both the training speed and segmentation performance of the model. This paper proposes a novel FDL approach for reliable and efficient multi-institutional COVID-19 segmentation, called MIC-Net. MIC-Net consists of three main building modules: the down-sampler, context enrichment (CE) module, and up-sampler. The down-sampler was designed to effectively learn both local and global representations from input CT scans by combining the advantages of lightweight convolutional and attention modules. The contextual enrichment (CE) module is introduced to enable the network to capture the contextual representation that can be later exploited to enrich the semantic knowledge of the up-sampler through skip connections. To further tackle the inter-site heterogeneity within the model, the approach uses an adaptive and switchable normalization (ASN) to adaptively choose the best normalization strategy according to the underlying data. A novel federated periodic selection protocol (FED-PCS) is proposed to fairly select the training participants according to their resource state, data quality, and loss of a local model. The results of an experimental evaluation of MIC-Net on three publicly available data sets show its robust performance, with an average dice score of 88.90% and an average surface dice of 87.53%.