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The strong few-shot in-context learning capability of large pre-trained language models (PLMs) such as GPT-3 is highly appealing for application domains such as biomedicine, which feature high and diverse demands of language technologies but also high data annotation costs. In this paper, we present the first systematic and comprehensive study to compare the few-shot performance of GPT-3 in-context learning with fine-tuning smaller (i.e., BERT-sized) PLMs on two highly representative biomedical information extraction tasks, named entity recognition and relation extraction. We follow the true few-shot setting to avoid overestimating models' few-shot performance by model selection over a large validation set. We also optimize GPT-3's performance with known techniques such as contextual calibration and dynamic in-context example retrieval. However, our results show that GPT-3 still significantly underperforms compared to simply fine-tuning a smaller PLM. In addition, GPT-3 in-context learning also yields smaller gains in accuracy when more training data becomes available. Our in-depth analyses further reveal issues of the in-context learning setting that may be detrimental to information extraction tasks in general. Given the high cost of experimenting with GPT-3, we hope our study provides guidance for biomedical researchers and practitioners towards more promising directions such as fine-tuning small PLMs.