At least eight people were also abducted from the National Tuberculosis and Leprosy Centre in Zaria early on Sunday morning.
Two nurses and a 12-month-old child were among those seized, said a hospital worker.
There has been a recent spate of abductions from schools and universities for ransom.
On Monday, reports emerged of another mass kidnapping from a school near Kaduna city, about 80km (50 miles) south-west of Zaria.
The mother of a 15-year-old girl who was kidnapped from Bethel Baptist School told the BBC that 140 schoolchildren had been seized by a large group of armed men who arrived on motorbikes and broke down the fence.
In a statement, police said gunmen “overpowered the school’s security guards and made their way into the students’ hostel where they abducted an unspecified number of students into the forest”.
A total of 26 people – including a female teacher – had been rescued, the statement said.
A local Christian leader said there were 180 students in the school, only 20 of whom had been accounted for so far. However, he said some of them may have escaped.
Police said gunmen involved in the hospital attack in Zaria, thought to be from local criminal groups, opened fire on a police station in the city. While they were engaged in the shootout, another group attacked the hospital.
“The attack on the police station was a distraction whilst another group attacked the dormitories of the health centre workers,” a local resident told AFP news agency.
The group escaped with the victims into a nearby forest.
A hospital worker, who asked not to be named, told BBC Hausa that the gunmen had abducted at least 12 people, including three children under the age of three and a teenager.
A local government official said troops were stepping up efforts to find the victims.
More than 1,000 students have been taken since December and nine have been killed. More than 200 students are still missing, some of them as young as three.
Authorities say recent attacks on schools in the north-west have been carried out by bandits, a loose term for kidnappers, armed robbers, cattle rustlers and other armed militia operating in the region who are largely motivated by money.
Since the well-publicised abduction in 2014 of 276 schoolgirls from Chibok secondary school by Boko Haram Islamist militants in Borno state, more armed groups have resorted to mass abduction of students.