This is a terribly general and incomplete question looking for an impossibly general answer.
Before I attempt to paint a scenario or two as answers, I need to draw your attention to what have not been stated in your question as they could lead to dramatically different outcomes. If you had a particular business in mind, then it would be best to paint a full picture while leaving out all the identifiable bits.
It all depends on so many things you have not said, not least on what you actually meant by "finances" which I am interpreting as cash balance, and what you mean by no growth which I take as revenue growth. In fact, if the market does not change and the business is already reinvesting a portion of its cash flow as "maintenance capex", then there is a pretty good chance that for at least some duration accounting profit would rise as some of the assets are fully depreciated. With that, tax would go up. While profit is just a number, tax is tangible cash.
It also depends on the state of its balance sheet; just because it is "very profitable", you cannot assume that it is not excessively leveraged. Taking everything as you had described, this company's "finances" will continue on the same trend, up, down or level.
You have not said what kind of sector this is in and what kind of risks it is exposed to.
The company is "very profitable" and with no prospect of revenue growth, owners quite rightly expect a high payout ratio. It would also be adding economic value. If it is optimally leveraged for now, then the value to equity owners would be naturally magnified. If not, it would still be adding value. It can pay out all its current and accumulated profits and to its equity owners, this would be just an annuity with some risk. And its value would likely be driven by dividend yield.
You have not said anything about what kind of free-cash flow profile is expected. Does it need significant reinvestment to sustain this "very profitable" business? If there is no need for cash call, why would share price matter other than the board's responsibility to maximise shareholder value? Surely long term "finances" cannot be predicted without first addressing short term corporate governance. At the AGM, surely the board will be confronted on massive pay rises - it is hard to imagine management (and directors) not participating in this hike.
A lot will depend also on previous and future cash retention and what the payout ratio has been against accounting profit and more importantly against operating cash flow. So, "finances" could be in decline, just the same or improving - only you can supply the missing pieces to complete the scenario choice.
With all the behind us, we can look at a market value erosion. If there was a market revolt, it is likely to overreact. So, its intrinsic value may well be above market value, perhaps by a long margin. In time, liquidity of its shares would likely fall. Any selling (or buying) could exaggerate price movements. For a no growth business at below intrinsic value, is that not a great justification for a management buy-out? Or for a activist investor to take a meaningful stake, force a board and management change to reinstate better payout and reap the capital gain from market value restoration? If I were management, surely it might even make greed-driven sense to take on debt, make a leveraged buy-out and make a double gain from the higher payroll as well as take ownership of this annuity of a business at a discount? At that point, the lines between dividend and payroll becomes a little blurry. In fact, this kind of behaviour is not uncommon at all.
As written, your question is really not about finances, nothing will change in finances just because market value of equity drops. The real question is what could happen to the company's ownership and/or ownership structure.