Quick answer: There is an infinite copper vein right next to your base — in the rock face west of your Yard, by the pond where you first fish (orange flecks in the rock). Mine it with a pickaxe (mastery 2); each stage gives 3 Copper Nuggets and it never runs out. Pan tin from river sand at the Sluice Box: 3 copper + 1 tin = 5 bronze ingots.
This page covers renewable copper source (source renouvelable de cuivre) and polished copper.
Steps
Get a pickaxe with mastery 2
A bronze pickaxe is enough.
Mine the vein
In the 100% run: "Did I mention there's an infinite copper node right next to my base?"
Dig sand by the river
Run it through the Sluice Box to get tin.
Smelt bronze
3 copper + 1 tin = 5 ingots. That keeps you in Bronze Nails and Polished Bronze once the barrels run dry.
Polished copper
The polished copper recipe is not confirmed yet; this section will be updated once it is verified in game.
Sources: a recorded 100% playthrough on YouTube, Game Rant, Prima Games.