Privacy preservation calls for speech anonymization methods which hide the speaker's identity while minimizing the impact on downstream tasks such as automatic speech recognition (ASR) training or decoding. In the recent VoicePrivacy 2020 Challenge, several anonymization methods have been proposed to transform speech utterances in a way that preserves their verbal and prosodic contents while reducing the accuracy of a speaker verification system. In this paper, we propose to further increase the privacy achieved by such methods by segmenting the utterances into shorter slices. We show that our approach has two major impacts on privacy. First, it reduces the accuracy of speaker verification with respect to unsegmented utterances. Second, it also reduces the amount of personal information that can be extracted from the verbal content, in a way that cannot easily be reversed by an attacker. We also show that it is possible to train an ASR system from anonymized speech slices with negligible impact on the word error rate.