Is there a way to update WordPress without the use of FTP, but with SFTP (which uses SSH)? I’ve only got my server set up for SSH access (and therefore SCP/SFTP).

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Configuration instructions in Codex on Enabling SSH Upgrade Access recommend either:
- Using SSH SFTP Updater Support plugin from official repository.
- Using built–in SSH2 support, which requires the PECL SSH2 extension installed on the server.
It used to refer to this tutorial: Using SSH to Install/Upgrade in before, which might be of use.