Yes. Open source software is copyrighted, and its license works only because of that copyright: the author keeps ownership and uses an open source license to give everyone permission to use, change, and share the code.
The Open Source Initiative (OSI), the organization that approves open source licenses, explains that a work must still be in copyright to be subject to a license. That is why open source projects carry copyright notices, and the OSI says you should never strip the copyright notices out of open source code and put in your own.
Contributors usually keep the copyright on their own changes. Under a contributor license agreement, a contributor keeps ownership but grants the project broad rights to use the work, while a copyright assignment agreement transfers ownership to the project. Software placed in the public domain is a different case. The OSI recommends applying an approved open source license instead of trying to waive copyright, partly because such waivers are not possible in every country.
Source: Open Source Initiative, Frequently Asked Questions
Image: Code on a computer monitor, Markus Spiske, CC0, via Wikimedia Commons.