Is there any kind of ear infection (outer/middle/inner) which is painless?

Depends: So, acute (recent onset) infections of the middle and outer ear are definitely painful. They can, however, transition and become ongoing processes which are dangerous despite not being painful. Most inner ear infections are not painful. Regardless, don't let the pain go untreated. See an ENT who can competently evaluate your ear and give you good advice/treatment.