No, the employment contract does not have to be in writing.

A contract means two people have agreed to something. There is an employment
contract if an employee shows up at work, does the work, and is paid by
the employer. It is important that both the employer and the employee agree
on what the employee is to do, and how much the employee will be paid.

However, a written contract helps if there is a disagreement about the
contract. It is easier to prove what the employer and employee agreed
to if you have a written contract.