You need a mobile P3 - NOT a mobile P3-m or earlier MMC-2 P3.

Are you sure you got a i4000/C600 motherboard and NOT a i4100/C610 model. The CPU from a i4000/C600 will NOT work in a i4100/C610 motherboard, different CPU although both are P3's and some of the same speed.

I suspect you got a i4100/C610 board as either model/revision of a i4000/C600 board will support the 850MHz CPU.

The older CPU is fine.  The bios check throws an error about "this CPU is not supported by this motherboard version" or something along these lines.  I can check when I get home, and post again.  So I really don't think 

One sure way to tell is the i4000/C600 does not have a onboard network card and doesn't have a spot for a modem daughter card. So the wires coming from the network and modem jacks on the side should go to the MiniPCI slot area and just end with connectors/plugs and they should be right next to one another. Also the i4100/C610 model have a rear fan where as the C600/i4000 doesn't so if there is a open socket at the middle back of the board that looks like a power socket that is for the rear fan. If the board you have has one of those sockets or has a spot for a modem daughter card and the wires coming from the NIC socket go to the board itself then that is a i4100/C610 board. I looked at the link you posted I can't tell from that picture.

Inspiron 4000 BIOS versions A06, A10 and A12 added CPU support for later Pentium III speeds; perhaps your BIOS should be Updated on the board. The trouble in that case would be that you need an older, slower CPU to do the UPdate, and then trade out for your 850mhz afterwards.