From Longman Dictionary of Contemporary Englishbe up to your neck in somethingbe up to your neck in somethinga) BUSY/HAVE A LOT TO DOto be very busy with something She’s up to her neck in work. b) PROBLEMto be in a difficult situation that is hard to escape from Jim’s up to his neck in debt. → neck
Examples from the Corpus
be up to your neck in something• Like Patsy Kensit, I was up to my neck in oasis.• The party is up to its neck in a scandal over alleged illegal purloining of confidential police files on rivals.• We were up to our necks in problems with the Apollo program. 