Does carpal tunnel of the feet exist?

Sort of: Carpal tunnel syndrome is a median nerve entrapment at the wrist. A similar condition can occur in the foot/ankle with a tibial nerve entrapment in the tarsal tunnel and is not surprisingly named tarsal tunnel syndrome. Both types of patients can experience pain and numbness.

Tarsal tunnel: Carpal tunnel syndrome is strictly defined as entrapment of the median nerve at the wrist. There are some entrapment neuropathies of the feet, however, such as tarsal tunnel syndrome, that are seen in practice.

Yes: Its called tarsal tunnel syndrome.