The arrest comes as part of a long-running investigation following a high number of baby deaths at the hospital.
Detectives launched an investigation in May last year, initially looking at the deaths of 15 babies between June 2015 and June 2016.
On Tuesday police said they had made an arrest and the probe had widened.
Detectives said they are looking into the deaths of 17 babies and 15 non-fatal collapses between March 2015 and July 2016.
Police have not said if the arrested woman is a nurse, doctor or other health professional.
Detective Inspector Paul Hughes, in charge of the investigation for Cheshire Police, said: “The investigation into the neonatal unit at the Countess of Chester Hospital is continuing.
“Since it was first launched in May 2017, a dedicated team of detectives have been working tirelessly on this case.
“This is a highly complex and very sensitive investigation and, as you can appreciate, we need to ensure we do everything we possibly can to try to establish in detail what has led to these baby deaths and collapses.
“When the investigation was first launched it was focusing on the deaths of 15 babies that occurred between the period of June 2015 and June 2016.
“In addition the investigation was also conducting a review of six non-fatal collapses during the same period.
“Since the start of our inquiries and as the information gathering process has continued, the scope of the investigation has now widened. We are now currently investigating the deaths of 17 babies and 15 non-fatal collapses between the period of March 2015 and July 2016.
“Due to the nature of the case and as part of our ongoing inquiries, we have consulted with a number of medical specialists to ensure that we carry out as thorough an investigation as possible. We have also spoken to a large number of people to gather as much information as we can.
“As a result of our ongoing inquiries we have today arrested a healthcare professional in connection with the investigation.
“She was arrested earlier this morning on suspicion of murder in relation to eight of the babies and attempted murder in relation to six of the babies and is currently in custody.
“Whilst this is a significant step forward in our inquiries it is important to remember that the investigation is very much active and ongoing at this stage.
“There are no set timescales for this coming to a conclusion but we remain committed to carrying out a thorough investigation as soon as possible.
“We recognise that this investigation has a huge impact on all of the families, staff and patients at the hospital as well as members of the public.
“Parents of all the babies are continuing to be kept fully updated and are being supported throughout the process by specially trained officers.”
The hospital carried out a number of independent expert medical reviews into the deaths before calling in police.
Medical director Ian Harvey said: “We are continuing to support Cheshire Police with their ongoing investigation.
“Asking the police to look into this was not something we did lightly, but we need to do everything we can to understand what has happened here and get the answers we and the families so desperately want.
“The Countess is now equivalent to a Level 1 special care baby unit and we are confident the unit is safe to continue in its current form.”
A report published in July 2016 by the Royal College of Paediatrics and Child Health (RCPCH) found significant gaps in medical and nursing rotas and insufficient staffing for the provision of longer term high dependency and some intensive care.
The RCPCH report said paediatric consultants had been concerned since June 2015 about a higher than usual number of deaths on the ward, several of which were “unexplained” or “unexpected”.
The hospital appeared as busy as ever on Tuesday with dozens of patients, staff and visitors coming and going from the vast site in Chester.
However security guards were on patrol and stationed near hospital entrances as a growing media pack arrived close to the main entrance.
Hospital officials said they will not be making any further statements.
A neighbour said a woman lived at the address who moved in around 2015 but did not know her name.
She added: “She’s a bit of an enigma, no-one seems to know her.
“I saw lots of police cars arrive, a few cars then a big police van, and gradually more and more came and they put up the tent.”
Neighbours said police first appeared at the property at around 7am on Tuesday.
Officers moved a small, silver Suzuki car from the driveway of the address to erect the tent.
The car has an NHS parking permit in the window.