There are a few reasons why a firm might be technically profitable yet illiquid - the most likely reason is due to the fact that the business may be making a profit from fixed assets, yet has a lot of overheads. These overheads mean that the company, securities or assets cannot be sold very quickly , making them illiquid.
I'll try to explain in greater detail below:
For the purpose of this example, let's create a fake firm. We'll call this firm The Rental Guys.
Now, The Rental Guys have loads of properties that they're successfully renting out to tenants. In this instance, The Rental Guys is a profitable firm.
The issue at hand is that The Rental Guys are still paying for mortgages and overheads on their many properties. The Rental Guys, therefore, have debts that are a large burden on their successful profits.
In this instance, The Rental Guys firm is both profitable and illiquid, because the properties they own cant be turned into real cash very quickly due to the overheads and complexity.