Chile secured back-to-back Copa titles after a second consecutive penalty shootout win over Argentina in the decider.
However, Sanchez went off in the first half of extra-time after failing to shrug off an injury suffered in the opening minutes, courtesy of a crunching Gabriel Mercado challenge.
The Arsenal attacker posted an update of his swollen ankle, which features some heavy bruising, via his official Twitter account on Thursday.
Sanchez wrote: "Fourth day of my holidays, stuck in a hotel bored... but I am much better."
He then added: "Your body can withstand almost anything.. it's your mind that you have to convince."
Once recovered, Sanchez will link up with his Arsenal team-mates for pre-season.
It is unsure whether he will be fit for their first game against an MLS All Star team on 28 July.
Arsenal's opening Premier League fixture is against Liverpool on 14 August.