Does constant money supply hinder economy? Consider an hypothetical isolated economy (an island) where:


the only form of money is Coins
the amount of Coins is constant
the Coins are never lost or destroyed


In that scenario:


can the economy grow smoothly (e.g. produce more goods, produce at lower prices, develop new goods, etc.)? 
or does the Government of the island need to increase/decrease the amount of Coins from time to time, to ease the economic growth?


I suspect the second is true but the mechanisms are not clear to me. Help!
The second is true. The reason why the second is true are macroeconomic friction and the fact that constant money supply in presence of growth would lead to monetary instability.

The mechanism works as follows; If you fix money supply and your economy grows you will get deflation as you can see from the standard monetary equation $MV=PY$ where $M$ is money supply, $V$ velocity of money, $P$ aggregate prices and $Y$ output. Since the M is fixed by assumption and V does not usually change much, any economic growth (increase in Y) must be accompanied by deflation (decrease in P).

Now depending on how high the deflation is this would be bad in itself. Even if there would be no macroeconomic frictions deflation similarly to inflation is form of price instability and even frictionless economy would be able to handle only little bit of it. In presence of deflation people will unnecessary hoard money instead of actually using it in the economy, it redistributes wealth from debtors to creditors and so on. Hence, even in completely frictionless world it could be proven that the optimal monetary policy should be such that change in aggregate prices is exactly 0.

However, unfortunately we are not living in frictionless economy. In real life there are sources of frictions which make deflation dangerous. There are several of them but to keep this concise I will focus only one of the biggest one sticky wages.

Sticky wages mean that companies cannot change their wage costs in short run. This is because usually contracts have fixed wage and to change them you need to renegotiate them. If wages are sticky in presence of deflation firms that can’t decrease their wages quickly enough (and workers usually oppose pay decreases - even nominal ones very hard) will see their labor costs significantly increasing forcing them to unnecessary lay off workers thereby creating  unemployment that is higher then socially optimal. Unnecessarily high unemployment has its toll on economic growth especially when it leads to some people being permanently unemployed. Due to sticky wages and other frictions economists are usually quite vary of deflation and hence predominant mainstream view is that countries should run inflation of about 2% on average, by growing their money supply in ways that achieves that target.

An example that shows that deflation can really have serious bite is the Great Depression. As famously proven by none other than Milton Friedman and Anna Schwartz in their monetary history of US, Great Depression at the beginning seemed like a relatively mild recession, however the recession lead to contraction in money supply due to bank failures and consequently to high deflation. Instead of doing anything about it Fed was completely passive, and what according to Friedman and Schwartz could be short recession became the worst US recession in recorded history (excluding maybe the current covid recession which might beat it) and this was mainly due to extreme deflationary pressure that lead to excessive unemployment and exacerbated the negative demand shock.