A law coming to Finland will give the country’s data protection chief new powers to fine corporations for breaches and noncompliance and strengthen children’s data security.

The law, which kicks in Jan. 1, will help enact the European Union’s General Data Protection Regulation, which took effect in May. A lengthy debate on the tough sanctions mandated by the EU regulation delayed the law’s adoption, Finnish attorneys told Bloomberg Law.

“The two most contested points were related to fines—firstly, who should...