Looks like a bignum bug to me. It's certainly not the behaviour I would expect, though I haven't checked to see what (if anything) the bignum documentation has to say on the matter.
Update:Aaah ... but lamprecht has :-)

Safest workaround would be to explicitly use Math::BigFloat:

use strict;
use warnings;

use Math::BigFloat;
Math::BigFloat->accuracy(200);

my $s = sqrt(Math::BigFloat->new(2));
print $s, "\n";

for my $num (2..2) {
    $s = sqrt(Math::BigFloat->new($num));
    print $s;
}
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Cheers,
Rob