To cover or fill with water in the manner of swimming, or to swim throughout something.
From prefix 'be-' + 'swim' (from Old English 'swimman'). This rare verb would mean to thoroughly saturate with water as one would pass through it swimming.
This is an extremely rare word, but it shows the 'be-' pattern clearly — just as 'besmirch' means 'to thoroughly smirch,' 'beswim' would theoretically mean 'to thoroughly swim through' or 'to fill with swimming water.'
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