She arrives by 8 most nights, shuffling past City Hall plaza, her backpack in tow with her orange and black blankets, a cardboard mat, a change of socks, and a pair of fleece-lined slippers, all she needs even on the coldest rainy nights.

Settling down on the hard, concrete steps of a downtown police station, the small, plump woman nestles into her blankets, twitches and shakes when she tries to sleep. Patrol officers step over and around her as they bound out the doors on their way to the nightly medley of emergencies.

They are hardly oblivious to each other, the police and the woman named Diana who has made the station her home. In fact, it is quite the opposite. Officers accustomed to a world of shootings, robberies, drugs, and tragedy reach out to her, offering food left over from some community event. Others tell her to go inside. Some ask about her story and offer help.

“Each and every officer tries to keep an eye on her,’’ said police Sergeant Tom Lema, a 27-year veteran based at the A-1 station on New Sudbury Street.

Loitering, of course, is illegal, and rampant chronic homelessness is considered by many to be a blight on downtown Boston. But Diana’s nightly presence on the precinct steps offers an unusual glimpse of the small gestures of kindness that divide police work and law enforcement.

Police officers would rather see her there than learn something worse has happened.

“The thing is, you know where she is every night, and no one’s saying, ‘Where’s Diana,’ ’’ Lema, a community liaison to the city’s homeless, said recently in a way that reveals his passion for helping anyone on the streets. “You don’t want to push someone back on the street and worry she’ll become a victim.’’

Diana, 43, has a nest for hair and a gaping smile. She suffers from the darkness of a mental illness that blinds her from knowing she doesn’t have to sleep on the stairs. And her choice to do so reflects the challenges in helping the homeless, as Diana refuses to go to a shelter, and no one, not even police, can force her to seek help. But the result is a relationship between Diana and her protectors, a type not taught in police academies.

The A-1 district knows all too well the plight of the homeless and the mentally ill, particularly during the winter when temperatures drop and shelters are filled. Surrounded by most of the city’s main homeless programs, and home to Boston’s tourist destinations, where vagrants tend to panhandle, the district has a workload that mostly involves helping the poor and needy.