Prosperity but Not Perfection
At the risk of stating the obvious, the absence of an income tax has helped make the Cayman Islands very prosperous, 14th-richest in the world according to the latest data from the World Bank on per-capita economic output (top ten in the world if you exclude oil-rich jurisdictions).
This does not mean, incidentally, that economic policy is perfect in the Cayman Islands.
There is an overly large and excessively compensated government bureaucracy. Indeed, financing the civil service is becoming such a burden that the Cayman Islands almost made a suicidal decision to impose an income tax earlier this decade.