In short, no. DHCP is not designed to handle multiple entities on a single network segment. Likewise, if one of the other businesses were to set up a DHCP server, there would be nothing to prevent your company's computers from accepting IP addresses from another DHCP server. Because of the nature of DHCP, a DHCP client will accept an IP address from the first DHCP server that answers.
Who manages the building backbone? You may want to talk to them about segregating your company into a seperate subnet. You would probably need to provide additional equipment such as a router and hubs to do so.
Your only other route, barring getting the IT folks from the other companies to not connect a computer to the backbone until TCP/IP has been configured, and barring getting the building managment to allow you to segregate your segment from the rest of the building, is to use fixed IP addresses.