Can Market Makers Be Illiquid?

I am just learning about what market makers are and how they facilitate markets in sock exchanges.

I was wondering is there a situation where market markers cannot make markets? I was thinking, the scenario where there is no one willing to sell a share to start with. In such a case, I do not think there is nothing more the market makers can do…right?

Is that a valid situation that can happen in real life? If so what is done to fix it?