It comes after West Yorkshire Police chiefs warned that they were facing increasing calls on their time despite dwindling resources.
"Police are often the first point of contact for people in crisis and they have specific powers under the Mental Health Act to detain very unwell people under section". Figures from forces show 97,796 crimes and 431,060 incidents were flagged as involving mental health concerns in the year to June 2017, but inspectors said this was just the tip of the iceberg.
Officers have also said that some mental-health services advise people with problems to call police after hours because they have nurses who can offer help and treatment.
Police are having to ignore crimes to look after vulnerable people as they "pick up the pieces" of a "broken mental health system", a damning report has found. Officers are being swamped by calls about mental illness, with the country's biggest force receiving one such call every four minutes, inspectors said.
But John Apter, chairman of the Police Federation, which represents rank-and-file officers, said Ms Billingham's frank assessment "reflects the reality of what we are experiencing every day".
Mind was involved with the report - "Policing and Mental Health - Picking Up the Pieces" - by being part of the reference group.
"As a Force we have been vocal about changing the way people with mental health issues are treated".
Previous year just five people alone made an incredible 8,655 calls to the Metropolitan Police, which cost £70,000 ($89,100) to answer.
Its analysis found that the peak time for calls to police for support with mental health-related incidents is between 3pm and 6pm Monday to Friday.
Examples of police stepping in to fill shortfalls in health services could include transporting someone to hospital when an ambulance is not available or waiting at hospital until a mental health place is found.
She added: "All too often, the system is failing people when they most need help".
"Although police officers generally do a good job in identifying and responding to those with mental health problems, they must never be considered a substitute for expertly trained healthcare professionals".
"We can not expect police to pick up the pieces of a broken mental health system".