Your best bet is probably to search the Tapeheads forum. They have a whole subforum about repair services. Klassik can't say for sure if anything mentioned here is worthwhile, but the information here might be a start:
Approx. 3 years ago I had my Sony Walkman Pro repaired at Wilkinson's Hifi at Nelson Lancashire. Their websire gives contact details which includes tel. no. (01282 612901) and email, email@example.com
The downside is, because of his great knowledge, skills and reasonable prices, Andrew is so busy and therefore waiting times can be a little extended.
Seemingly a chip in the Walkman, which was the cause of many failures, was at one time obsolete and therefore unobtainable. Fortunately they are now being made again.
The Walkman Pro WM-D6C is an amazing player and has replaced a Nakamichi Dragon in my system. Not quite as good but nevertheless full featured and superb and it takes up far less space.
Because the gentleman in question is not a dealer, does not do these repairs for a living, and, whilst it is easier to contact him by telephone, I would prefer not to share his direct personal contact details on the open forum.
However, if you insist, he has a website all about Sony Walkmans and it has a contact page...