Why?
This doesn't sound obviously true to me. Imagine an economy in perpetual long-term stasis, with zero inflation. There's a company which has a factory which turns a profit, because it makes useful stuff. Every ten years, they need to replace the factory. So they take out a loan, and pay it off, with interest, over ten years. The lender earns interest with positive real value from this. Is there anything in this story which is impossible? Is there a theoretical reason a world like that couldn't exist?
Let's expand on the example of your factory. Say that you have a factory, that needs to be replaced in ten years. Why would you take out a loan over ten years with substantial interest, when you can simply save up over ten years to buy another factory? Since there is zero inflation, this is incredibly better for you. And so, there is no reason for you to take a loan; and you won't take out a loan. For an economic transaction to happen, it needs to be in the interest of both party, and in a no-growth economy taking out a loan is either in the interest of the lender or the debtor, but never both, on average, for a given lending policy. This is not the case when there is growth, which is why we have banks, whereas we didn't really have any kind of lending infrastructure in times were growth was of or near zero.
The only people that might want a loan are people that don't have enough capital to get started, but in a zero-growth economy you cannot expect that they will be able to repay you, so you won't lend to them, and if you do lend to them expecting that they will not pay you back that will be at astronomical interest rates and will essentially be usury.
Thus, growth must always exist for the current system to continue.