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A couple of options. As you've noted, the airport doesn't have luggage storage. Hotels/hostels often will store for a fee, or worst case, book a 'night' there even if you don't use it, and just 'check in' and store your bag.
Flying Dog Hostel Iquitos is one example that offers luggage storage:
And according to Wikitravel, the Emperador Terraza Hotel also offers this service.