Sony Walkman WM-D6C
If your WM-D6C has a serial number below 267201 (take out the battery block and you'll see the serial number sticker there) then the problem is electronic, not mechanical. It is likely due to the speed controller chip (labeled as IC601 on the circuit board), which is made by Sony and is type CX20084. This chip decides the voltage that is fed to the motor which is only about 2.4 volts, Should the chip be defective then the motor receives about twice this voltage and thus runs too fast. If you take the back off the recorder you will see the circuit board; it looks like a rectangle with a small square extra bit sticking out to the right. It is here that the IC601 resides. It is about the size of a small pinky finger nail and has 8 tiny legs down each side. Measure the voltage on pin 15 (put one voltmeter probe (+) on the second leg down on the right side (pin 15) and the other one (-) on the battery negative), it should read about 4.8 volts, which it feeds into the base of transistor (Q601) that in turn will then feed 2.4 volts to the motor (if everything is working OK). If the chip is damaged, then the voltage on pin 15 will be close to zero, and the transistor Q601 will not be "held back" and so will feed about 5 volts to the motor. CX20084 is (unfortunately) obsolete, someone in the UK sells them for about $50, but an excellent company in Hong Kong (utsource.net) sells them for $5 (plus $8 S&H).
Taking out the old chip and putting in a new one is close to microsurgery, you will need a micro soldering iron (18Watt Antex), a good eye, a powerful magnifying glass and a steady hand, but it can be done - I've done it!
There is a possible second reason that could cause the motor to run too fast. There is a pick up coil on the main flywheel called FG901. If this is damaged it will be open circuit and then the only use for your recorder is probably a paper weight. You can test this by going back to chip IC601 and locating pin 1 (top left pin; a small circle on the body of the chip is next to pin 1), Pin 1 is connected to the positive side of a capacitor, C602 (0.47 microF), check the resistance from the negative side of C602 to the ground (take battery out, and use battery (-) as ground). You should get about 600 - 650 ohms reading. If you get nothing, sorry there is not much you can do about it as these pick up coils are definitely not available.
Finally, chip 601 (CX20084) goes bad if you connect the batteries back to front. Good luck.
Giles..