Thai lacks grammatical gender, but men and women use different first person pronouns. Actually it’s more complicated than that, there are a number of ways to refer to oneself depending on context and the relative status of speaker and addressee. But the choices are different for men and women, generally. For example, women who want to sound polite may need to refer to themselves as ‘mouse’ when talking to higher status people. A linguistic study of this by Diller and Chirasombutti found that “Thai women are required by the prevailing linguistic system to ‘place themselves’ through self-reference selections in a more finely-determined social space than that required of male speakers.”
This also correlates with sentence-final polite particles (men use khrap, women kha). It’s such a thorough-going system that it would be extremely difficult to change – getting rid of it would be a radical alteration in the ways politeness is communicated. On the other hand, you could say it’s PC even though not gender-neutral, because it can accommodate transgender people. E.g. a biological male who identifies as a woman can use feminine pronouns/particles to assert his/her gender identity.
According to Dixon (Basic Linguistic Theory vol 2, p. 200), “very many languages” do distinguish gender in third person singular pronouns only, so you may be able to find some without grammatical gender that do so. But be aware that there are many other possibilities cross-linguistically, as with Thai.
Reference: A. Diller and V. Chirasombutti, "Who am ‘I’ in Thai? – The Thai First Person: Self-reference or Gendered Self?" in P. Jackson and N. Cook eds, Genders and Sexualities in Modern Thailand, (Silkworm 2000), pp. 114-133.