Can a wife complete breastfeeding her nine-month-old son while she is one month pregnant, given that there are Prophetic hadiths prohibiting that?

There is no prohibition against a pregnant woman breastfeeding. Rather, the Prophet (peace be upon him) intended to forbid it out of fear of harm, but he did not do so when he saw that the Persians and Romans did it without any harm befalling their children. "Gheelah" is when a man has intercourse with his wife while she is breastfeeding, or when a woman breastfeeds while she is pregnant. Scholars deem "gheelah" permissible because the Prophet (peace be upon him) did not forbid it. Ibn Uthaymeen stated that there is no sin in having children successively, even if the mother becomes pregnant and the previous child has not completed breastfeeding, indicating that breastfeeding a child during pregnancy is permissible unless there are medical reasons that necessitate its prevention, as determined by a physician.