Does Islam oppress women?

No. On the contrary, Islam elevated the status of women 1400 years ago by giving them the right to divorce, the right to have financial independence and support, and the right to be identified as dignified women (Hijab) when in the rest of the world, including Europe, women had no such rights. Women are equal to men in all acts of piety (Quran 33:32). 

Islam allows women to keep their maiden name after marriage, their earned money, and spend it as they wish, and ask men to be their protector. Prophet Muhammad (pbuh) told Muslim men, "the best among you is the one who is best to his family." Not Islam, but some Muslim men, do oppress women today. This is because of non-Islamic cultural habits or their ignorance towards their religion.