No. Every EU country is required to implement GDPR as a law. All government bodies have to abide their own countries laws.

As an example; the implementation of GDPR in The Netherlands is called AVG, and it's currently discussed that the tax authorities might be the first to receive a warning or fine.

This site has more info : https://www.eugdpr.org/gdpr-faqs.html

But I would say government offices and public sector organisations do fall under this:

".. It applies to all companies processing and holding the personal data of data subjects residing in the European Union, regardless of the company’s location."

No, they're not, in fact they often have a higher standard to meet (more likely to need a DPO, for example).

The government is mandated by law. GDPR clearly states that requirements set by law take precedence, doesn't matter what kind of organization or business.