You can stay based on the time the immigration gives you upon arrival. CBI application does not grant an automatic right to stay in the country. You need to confirm your current citizenship in order to determine what entry visa is required.
What are my options if I want to stay in St. Kitts and Nevis while waiting for my CBI application to be processed?
Is it possible to keep staying in St. Kitts and Nevis while my CBI application to be processed? Do I need to apply for a special type of visa? Does the fact that I have a pending CBI application automatically grant me the right of staying in the country?