The BLE application we have made is built on the zephyr RTOS. Because of this it's not possible to configure the system to go into deep sleep through the use of the application-level API used in the BLE samples. To allow zephyr to drive the power consumption, power-management functions need to be hooked into zephyr so that zephyr can use them.
Having looked into how the BLE samples put the system to sleep, it appears they use the same function as described earlier, being the POWER_EnterPowerMode function in fsl_power.c. This is the function that allows the system to be put into power down and deep power down mode, and in which the function that supports deep sleep is commented out. There are some functions in the BLE framework which have the word deepsleep in them, but they lead back to functions which put the system into power down or deep down. It appears to me that the samples don't put the system into deep sleep, so the BLE samples are a dead end.
I could try to write my own implementation, but the lowest-level functions for putting the system to sleep are undocumented functions stored in ROM. I could use the same ROM-functions used in putting the system in power down and deep power down, but this is very risky as I could be missing vital functions and information for deep sleep specifically.