Yes and no. For all practical purposes, the most popular and common implementation of SSH, OpenSSH, is free and open source, so yes. Where the "no" comes in here is more due to SSH's history.
It was initially freeware developed by Tatu Ylönen of SSH Communications Security, but later versions from SSH Communications Security were made more and more proprietary. This is about the same time that OpenSSH was developed, and its open nature has made it the de facto standard SSH server.