No—an ordinary polarizing filter should not have a small section that appears unpolarized. Whether it’s a linear polarizer or a circular polarizer, the polarization effect should be consistent across the whole filter.

The main exception would be a different kind of filter, such as a graduated neutral density filter, where part of the glass is intentionally clear or less dense. But that design does not apply to a standard polarizer.

Based on that, if part of your polarizer seems not to polarize at all, it’s most likely defective and worth returning or exchanging.