Trajan's Market was a sort-of shopping mall built in Rome 100-110 AD. It was a large building featuring several stories of shops and other establishments such as taverns, a library and a large hall believed to be used for audiences or concerts. It was commissioned by the emperor Trajan and designed by his trusty architect, Apollodorus of Damascus.

The bazaars of the east could also be considered a sort of shopping mall. The term bazaar originates from the pre-Islam Persian society. In the far east, the city of Kashgar boasts a bazaar more than 2000 years old. Situated in what is now the far western end of China, it was reportedly the largest market in Asia. It was first described by the Chinese explorer Zhang Qian in 120 BC, making it quite a bit older than Trajan's Market in Rome.

In my opinion, the modern shopping mall is nothing new. The idea of gathering a bunch of different shops in one location has probably been around as long as cities themselves. The modern, huge, multi-story version is simply the effect of people being better at building large houses these days.