The 486 was the first set of systems where the BIOS was changable on-chip; even then, you had to remove the chip, expose it to UV light, and reprogram the thing. Actual Flash ROM's in PCs didn't become widespread until late into the Pentium Era.
The Original IBM PC, AT-286 and AT-386 PCs implemented their ROMs via true-to-their-name ROM chips; high-end late model 386 PCs might have implemented their BIOS as EPROM chips (UV erasable) but EEPROM chips didn't become something to expect in a average PC until around 1995/1996.
I of course speak of PCs... since the 286 is really a processor, there are probably a lot of applications of the 286 being used in things like barcode scanners and so forth that do use a programmable ROM.... that being said it isn't strictly speaking a BIOS in that instance anyway.
Additionally, one must define what it means to "flash" the BIOS. "Flashing" the BIOS really sort of requires, well, a Flash BIOS. Flash BIOS chips use Flash Memory- EEPROMs- which didn't appear until 1995-96. the first programmable BIOS chips, the PROMs, arrived relatively early (these were present in 286 PCs) however it really didn't matter, you couldn't change what was on them anyway, they were only written to once. Just made production cheaper, since no longer did the actual circuitry itself contain the data, but rather a more abstract ability for manufacturers to "burn" the BIOS code onto the chip. Only once, though.
EPROM chips were (as I mentioned before) erasable with UV light over their little window, but you still needed a BIOS programming station in order to write a new one to it. Certainly not something you can do in-place.