If we are taking about a traditional zombie, the way they are often portrayed in movies (corpse raised by virus) then it's not possible.
An “engineered” zombie (corpse raised by some sort of nano machines) is theoretically possible.
A zombie like from the last of us (living person who’s behavior is changed) is possible.
A parasitic fungus that infects a unspected hosts and controls them with the goal of spreading more spores and infecting more hosts.
It already exists in nature. Cordyceps a fungus that infects ants and once an ant is infected, it loses control over it's body and force it to reach anything tall and burst open releasing the spore and infecting more ants. The fungus is literally called the “Zombie Fungus.”
There is also a parasite called Toxoplasma gondii that infects rats. Once a rat is infected it loses fear of cats, why cats? Because the only host where the parasite can reproduce is a cat.
Then the infected rat wonder around cats without any fear and gets eaten up and the parasite reproduces and continues the cycle.