Is it possible to run the code in external EEPROM
What code?
What chip, specifically, are you using? (you need to state manufacturer's full part number - including all prefix & suffix characters)
What, exactly, do you mean by, "EEPROM"?
Generally, EEPROM is used to store data that should be remembered even after power goes off. FLASH is a better alternative to store code.
Except that flash and EEPROM are almost the same thing - the main difference is the erase block size.
But in this case, the EEPROM must electrically connected and addressable as CODE for the processor to be able to run code out of it.
The initials, "EEPROM" stand for "Electrically-Erasable PROM" - and it is perfectly accurate to describe Flash as "Electrically-Erasable PROM"
Many people, however, take "EEPROM" to refer to a particular type of "Electrically-Erasable PROM" device specifically optimised for use in non-volatile data storage applications.
That is why I asked the OP what he means by EEPROM.
common (unclear) usage is to refer to serially accessible non-volatile memory as EEPROM and parallel accessible as flash.
nevertheless, I know of parallel EEPROM and serial Flash.
Erik