Yes, you can! Anguilla is a British Overseas Territory with 1 private school offering the IB education, however only the Primary Years Programme.

Primary Years Programme

The Primary Years Programme is available at _one _school:

• Omololu International School (The Quarter)
  Tuition fees: $5850-$7350

Middle Years Programme and Diploma Programme

These programs are not available anywhere on Anguilla. Sorry.

Sources:

• http://www.expat-quotes.com/guides/global/visas-and-permits/international-schools-in-anguilla.htm
• https://www.ibo.org/about-the-ib/the-ib-by-country/a/anguilla/