I have read and heard from several different sources that a professional translator can translate 3,000–3,500 words per day, as an average figure.
However, this range can vary a lot. I think this number of words is very doable in translating from one Romance language to another (Spanish to Italian) or from a Germanic language to a Romance language (English to Spanish), but I imagine this number of words might not be possible when translating between two very different languages (i.e. Arabic to English and vice versa).
I am able to comfortably translate 3,500 words a day, or even more, when I am working with colloquial or informal text. However, when working with technical or academic translations, I average a little less at somewhere between 2,000–3,000 words per day. Technical translating takes more time due to formatting considerations and field-specific terminology. I might be able to translate 6,000 words or more a day, but then I will need an additional day to edit or review this text. So overall, my translated words per day balance out to around 2,000–3,000.