Prime Minister Andrej Babis told Czech public television the shooting took place around 7 a.m. in a waiting room. The attacker opened fire at people's heads from close range, Babis said.
He said he was heading for the site, at the University hospital in the eastern city of Ostrava, 350 kilometers (220 miles) east of Prague.
Police said the man left in a silver-grey Renault Laguna car and called on the public not to try to stop him.
Police published a photo of the suspect, having withdrawn an earlier photo of a different man. They said that man was now considered to be a witness.
Video footage and pictures published by public radio showed police arresting a person at the site but that person did not appear to be the shooting suspect.
Officials say people have been evacuated from the clinic. Police are boosting security across the country.