Sophie Smith, 21, had been wearing only a vest and shorts when she left her home in Gorleston, near Great Yarmouth, Norfolk, at around 3am on 26 December.
The University of East Anglia student had been suffering from severe anxiety and depression and had been receiving treatment, police said.
She had also previously talked about ending her life in the sea.
Inspector Will Drummond said: “We know from speaking to her family that Sophie had previously spoken about her intention to enter the sea to take her own life.
“Sadly, we can also confirm that Sophie had left a note and messages for loved ones, outlining her intentions.”
Miss Smith had not taken her phone with her and she has not used her bank or social media accounts since disappearing.
In the only CCTV image of her that morning, she was seen running towards the beach at 3am.
Police and volunteers had been searching the coast for the young student.
Some 28,000 people joined a Facebook group dedicated to finding her.
Sophie’s mother Lynn Shaw said: “Sophie’s step-father Kevin and I would like to thank the public and police for their continued support and dedication in searching for Sophie.
“In our hearts we hold every hope she will return safe and well.
“However, as a family, knowing the troubles she was experiencing, we do fear the worst for our little girl.
“Sophie had openly discussed how she was struggling and was receiving treatment.
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“I would urge anyone suffering anxiety or depression not to fight that battle alone and seek help.”
:: Anyone feeling emotionally distressed or suicidal can call Samaritans for help on 116 123 or email email@example.com in the UK.