Are the Channel Islands Part of the United Kingdom?

No. The Channel Islands are Crown Dependencies—self-governing possessions of the British Crown. They comprise two separate jurisdictions: the Bailiwick of Jersey and the Bailiwick of Guernsey (which includes Alderney, Sark, Herm). The UK is responsible for defense and international relations, but the islands have their own legal systems, tax regimes, and legislatures (States Assembly in Jersey, States of Deliberation in Guernsey). They are not part of the UK, the EU, or the European Economic Area.