No, it would not, this is precluded by the EU treaties themselves (not some random regulation), would be completely unworkable and defeat the whole purpose of the single market. The EU is a customs union with a common external tariff.
Note that presently, tariffs really aren't that big of a problem. As many “hard Brexit” proponents have pointed out, WTO rules aren't that bad. What modern trade agreements (and the EU single market) are about are non-tariff restrictions, harmonising standards, services and intellectual property, reducing formalities, services (including posted workers and “passporting”), etc.