The Uffizi Palace in Florence (now an art museum) was built in 1560 as the offices for the Florentine magistrates. That is why it was named the Uffizi.

http://en.wikipedia.org/wiki/Uffizi

Coincidentally or not, the 1560s is also given as the date when the word "office" in English was first used to refer to "a place for conducting business". Before that the word was used to refer to a position or a post or employment to which certain duties were attached.

http://www.etymonline.com/index.php?term=office&allowed_in_frame=0

Do you mean in the sense of a dedicated office structure, or offices within a structure?

Because if the latter, then the answer is "as long as there have been political buildings". Even back in ancient Babylon there were administrative offices within the temples, and any palace has always had offices for the clerks and whatnot who did the actual business of running the government.

Even before the Japanese invented their own system of writing, I know that in Japan court functionaries and their staff had space dedicated to what we'd term office work.

If you mean "buildings which contain nothing but office space", I can't help much.

I think you should develop your question further. When you say "offices", what exactly do you mean?

Your definition of office does remain open to interpretation, however when it comes to banking, I believe I can offer some insight. In Paul Strathern's 'The Medici, Godfathers of the Renaissance', he discusses the opening of branches of the Medici bank across Italy.

'The new head of the Medici firm [Vieri Medici] was not interested in politics and devoted himself entirely to building up the business, opening exchange offices in Rome and Venice and conducting an export-import trade through the river port of Pisa' (Strathern, 2003; 22)

Strathern dates this as happening after 1388 with the death of the former head of the family, Salvestro. However he also mentions other banking families who operated in the same manner, such as the Bonsignori banking family who went bankrupt in 1298, so the concept of offices as such was familiar by this time period though my knowledge outside that of banking is limited.

Additionally, there were such advances one would associate with use of offices such as 'double entry bookkeeping [invented in 1340]... and payment by ledger transfers and bills of exchange' (Strathern, 2003; 19).