The 286 protected mode did not allow for a 32-bit flat address space and was heavily half-baked in other ways, e.g. no inbuilt way to return the CPU to real mode without a slow and fiddly CPU-reset.
It was architecturally a 16-bit CPU so a flat 32-bit address space would be a non sequitur. If you wanted flat 32-bit addressing, there was a contemporary chip that could do it with virtual memory: Motorola 68010 + the optional external MMU. (Or if you were willing to do some hoops, even a 68000.. see the Sun-1)
Protected mode on the 286 allowed 24-bit addressing, enabling access to 16 MB of memory, but lacked virtual memory and required a reboot to return to real mode. The 386 introduced virtual memory through paging, 32-bit addressing for 4 GB of memory, and virtual 8086 mode for running multiple 8086 programs simultaneously without compromising security.
