I have an ASRock Z390M-ITX/ac motherboard with a soldered/non-removable BIOS chip (this is apparently not uncommon on ASRock motherboards). I initially tried using test clips, but I wasn't able to get a good connection. Fortunately, the motherboard features a BIOS_PH header that is wired to the BIOS chip. Despite existing on multiple motherboards, its pinout is seemingly only documented in the J3355M manual ( http://asrock.pc.cdn.bitgravity.com/Manual/J3355M.pdf). Using this pinout, I was able to connect some jumpers from the BIOS_PH header to a CH341A programmer and successfully read from and write to the chip (a W25Q128JVSIQ in my case).
For reference, the BIOS_PH pinout (corresponding to a 25xx chip) is:
DI | GND CLK | no connection no pin | DO VCC | CS HOLD | WP