Hi, I have good news. I had the same problem when I've bought a used familyfix base. The reason for this non stop test mode is corrosion on the chip contacts due to liquids spelt onto the base. The trick is to have some star allen keys old toothbrush and rubbing alcohol, FIX: open the battery cover, batteries out, unscrew two tiny screws - very small star tip, flip over the schematic. Mine had white-blueish corrosion spots on the chip and some small capacitors/resistors. That probably short circuited the system into test mode. Grab your toothbrush, dip it in the cap of the rubbing alcohol filled with it. now gently rub the corrosion off. Along / parallel the contacts and the small circuit parts. Wipe it with a clean paper towel, repeat until it is all clean. Remaining alcohol will evaporate in a minute or two. Assemble back, batteries in, and it WORKS! Took me hours of research and i disassembled it of disparity. Good luck to all!
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On either side of the EasyFix base you will see a small square opening with the yellow IsoFix point showing through it. Take a flat head screwdriver and slide into this hole facing toward the front of the car and gently push the screwdriver handle so it ends up facing the rear of the car - this should unlock the IsoFix mechanism. You'll need to do it on both sides if both sides are stuck.
If this doesn't work, or once you have done this, take the base back to your retailer with your proof of purchase if it is under 2 years old and they will be able to assist.