The most common and most likely answer to this crossword clue is the 11 letter word PROGRAMMING
I've seen this crossword clue in The Times
P
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Buddy explains!

The straight part
We know that "brainwashing" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case brainwashing = programming.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does PROGRAMMING mean?
PROGRAMMING verb- To enter a program or other instructions into (a computer or other electronic device) to instruct it to do a particular task.
- To develop (software) by writing program code.
- To put together the schedule of an event.
- To cause to automatically behave in a particular way.
- The designing, scheduling or planning of a radio or television program/programme.
- Brain-washing
- The act of writing a computer program.
- The software that controls a machine, or the logic expressed in such software; operating instructions.