Are public schools considered 501(c)(3) nonprofit organizations?

The Internal Revenue Service (IRS) states the following regarding "instrumentalities" (including public schools) on page 16 of its Publication 557 under the heading "Section 501(c)(3) Organizations":

Instrumentalities: A state or municipal instrumentality may qualify under section 501(c)(3) if it is organized as a separate entity from the governmental unit that created it and if it otherwise meets the organizational and operational tests of section 501(c)(3). Examples of a qualifying instrumentality might include state schools, universities, or hospitals. However, if an organization is an integral part of the local government or possesses governmental powers, it does not qualify for exemption. A state or municipality itself does not qualify for exemption.

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