Explanation
Yes. Having six official languages does not prevent a UN General Assembly delegate from speaking another language. The delegation must provide interpretation or a written translation into an official language under the applicable arrangements.
UN Geneva's meeting guidance explicitly cites General Assembly rule 53. It explains that UN interpreters use the supplied interpretation or accepted written text to interpret the statement into other official languages. The educational publisher CliffsNotes independently describes the same allowance.
The six official languages are Arabic, Chinese, English, French, Russian and Spanish. Permission to deliver a speech in another language does not give that language official status. When a written translation is used, UN Geneva's guidance also requires someone from the delegation to help interpreters follow the speaker through the text.
Key takeaway
A non-official language can be used for a speech, provided the delegation supplies the necessary language support.