An EU student can work self-employed if they have pre-settled or settled status in the UK.
Any student holding a Tier 4/Student visa will not be able to work as self-employed or engage in business Activity.
Self-employment
A student visa does not allow self-employment. Self-employment normally includes activities such as freelance writing or publishing, private tutoring or selling goods or services directly to an end customer, for example as a consultant. If you are not on the employer or agency's employee payroll, it is likely the work being offered is on a self-employed basis. If you are unsure, it is important you check your employment status prior to starting work.
Postgraduate students undertaking undergraduate supervisions, or students undertaking other work in the university, must contact their programme lead to arrange a formal worker's agreement before starting the activity if they hold a student visa.
Engaging in business activity
A student visa does not permit you to engage in business activity. The Home Office defines this as working for a business in which you have a financial or other significant beneficial interest in a capacity other than as an employee. The Home Office provides some examples of the types of circumstances in which you would be considered to be engaging in business activity: setting up a business that is trading or has a trading presence; being employed by a company in which you hold shares of 10% or more; or working for a company where you hold a statutory role, such as a director. However, this is not an exhaustive list.
If you are unsure whether this restriction would apply in particular circumstances, you should seek further guidance from University of Plymouth
UKVI Compliance Team prior to undertaking the activity or email internationalstudentadvice@plymouth.ac.uk