Large-scale pre-trained language models such as BERT have brought much better performance to text classification. However, their large sizes can lead to sometimes prohibitively slow fine-tuning and inference. To alleviate this, various compression methods have been proposed; however, most of these methods solely consider reducing inference time, often ignoring significant increases in training time, and thus are even more resource consuming. In this article, we focus on lottery ticket extraction for the BERT architecture. Inspired by observations that representations at lower layers are often more useful for text classification, we propose that we can identify the winning ticket of BERT for binary text classification through adaptive truncation, i.e., a process that drops the top-k layers of the pre-trained model based on simple, fast computations. In this way, the cost for compressing and fine-tuning, as well as inference, can be vastly reduced. We present experiments on eight mainstream binary text classification datasets covering different input styles (i.e., single-text and text-pair), as well as different typical tasks (e.g., sentiment analysis, acceptability judgement, textual entailment, semantic similarity analysis and natural language inference). Compared with some strong baselines, our method saved 78.1% time and 31.7% memory on average, and up to 86.7 and 48% in extreme cases, respectively. We also saw good performance, often outperforming the original language model.