Hey Matej,
You should definitely have a sword once you speak to the old man. Did Armin say or do anything after the old man drops the sword? The intended behaviour is for armin to pick up the sword as part of the cut-scene and continue speaking for a while before the scene ends. I've personally never experienced what you're describing, so if you're able to recall what you saw as accurately as possible, that will help me try to figure out how to replicate and fix the bug.
I really appreciate your feedback, and I'm sorry you had that experience! The game would have been unplayable without a weapon beyond that point, which is a real bummer :/
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Nevermind, I was able to replicate it almost immediately. Patch incoming!