Particle board and water are not good friends. Trying to "repair" water damaged particle board is not particularly feasible and the real solution is replacement.
In your case the particle board has been covered with a hard laminate product called Formica (or other similar product) that is bonded to the particle board substrate with contact cement. If you were to just want to focus on replacing the back splash it would entail removing all of the old back splash and replacing with new. Doing so does however impose a number of potential big problems. Here are some of the considerations:
- The back splash of the countertop is likely built right into the top as an integral assembly of the substrates and then was covered with the Formica and trimmed in place with a laminate trimming router. Partial disassembly in this case would likely not be successful.
- If there is any age to this countertop it is unlikely that you can find matching replacement laminate material in the same pattern. The manufacturers of this material come out with dozens of new patterns and colors every year to keep pace with current trends and tastes. In addition the material can change looks over time and even new material of the same pattern may look obviously different.
- Even if the back splash can be individually removed it may become very difficult to install a new suitable substrate fastened to the wall and then having laminate installed in place. This would be due to how a laminate trimming router cannot be used on a narrow 1" top that is wedged up against the wall. If the replacement back splash was made up in the shop and then brought to the installation site there would be little choice but to try to glue it to the wall and the chances of getting a very good fit down to the existing counter top are highly unlikely. After all you would not want to see screws or nails through the back splash to hold it in place.
I can see two general repair strategies that may work for you. The first is to cut out the existing back splash and replace it with something contrasting and different. Tiles can be installed in a row across the back of the counter (in fact that is what my current 35 year old house has; laminate countertop with ceramic tile back splashes in the kitchen and laminate countertops in the bathrooms with red oak board back splash trimming).
The second strategy would be to remove all of the back splash and countertop materials and build in new replacements. A professional may very well suggest a replacement countertop material that comes pre-made with an integrated back splash and laminate that is continuous across the surface from the front edge all the way across and curved up and over the rounded edges of the back splash. If you want to retain the rectangular style like you currently have that can surely be built anew by a good craftsman but if you go that route consider using plywood as the substrate instead if particle board.
Covering a countertop with Formica laminate is a style that has really gone by the wayside in the recent decades in favor or other materials such as tile, stone, marble and other fake imitations of these. This may be a time to consider an upgrade to a more modern style.