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