Visas are not required of US citizens if their stays are less than 8 months. If you are looking for a longer-term solution, you may wish to apply for residency. One way to do that is to purchase property in the Bahamas meeting certain thresholds.
Annual residency cards are available to homeowners owning Bahamian Real Estate at any price point and can be renewed annually. The annual residency card allows the permit holder and their family to reside in The Bahamas for the duration of the permit.
Major international investors and owners of residences valued at $750,000 may apply for permanent residency.
At $1.5 million or more, owners will receive expedited consideration for permanent residency permits.
Applicants must be of good character, show evidence of financial support and commit in writing they wish to live permanently in the Bahamas. Spouses and dependent children (under 18 years) usually resident in a household may be endorsed on the certificate when the original application is made.
Source: Coldwell Banker Bahamas