Yes, of course you can keep your activity. Tax-wise, you should take advice from an accountant.
As a Golden Visa holder living in Greece, can I earn money from outside the country?
I am originally from Chile and still maintain a business there. Now, I live on Santorini on a Golden Visa from my property purchase. Am I allowed to earn income from my home country by working remotely in Greece? Do I have to declare any Greek taxes for this?
Provided that you have your residence in Santorini, Greece, you are considered as a tax resident of Greece, and subsequently you are obliged to declare here your annual global income, including the income that you earn from your business back in Chile.
The Golden Visa grants you the right to reside in a European country. You cannot work legally or make money from a third country because there will be a problem with tax collectors.
I understand how important the tax issue is for you and your activities, so I will try to give you some basic information about the existing legal framework. First of all, the permanent resident title for property owners does not mean automatically tax (fiscal) residency of the individual. The property owner can rent his property and the related income is subject to taxation, according to the existing tax scales in Greece. According to the law in force, an individual person is considered as tax resident of Greece, meaning that he has to declare his worldwide income in Greece, provided that: Greece is the permanent or principal residence, or habitual abode or center of vital interests, i.e., personal or economic or social ties; there is having a physical presence in Greece for a period exceeding 183 days within any 12-month period, continuously or intermittently, is a tax resident in Greece for tax year during which the 12-month period is completed. In any case it would be advisable to ask for tax consultation.