Browning makes a few O/U 20ga's with 26" barrels that will be around 6 lbs ..like the super light feather model that retails for around $ 2,000 -
$ 2,200 ....
And you don't say why you want such a light gun ...but just a note of caution, be careful that a "short and light gun" doesn't get real "whippy" and make you tend to "slap" at targets vs a smooth follow-thru...its all a trade-off.
There is another option ---go to a light semi-auto like the Benelli super sport in a 20ga ...28" barrel...and its right at 6 lbs as well ...and new its around
$ 1,800 ... and the 28" barrel on a semi-auto makes its overall length about the same as an O/U with 30" barrels ...so with the Benelli - you get a light gun / but a longer sight plane ...so again, its a nice compromise.
Everybody should buy what they want ....but this same weight in a 20ga came up at my club 2 weeks ago ...and a guy was convinced his wife needed a "light 20ga" ...in his mind, so it handled nimbly ...and it'd be perfect. When his wife came out and shot with us ...she hated the light 20ga that he'd borrowed for her to shoot...and since I had a Citori 20ga, with 30" barrels in my vehicle, I let her shoot it a little ...and she loved my Citori XS Skeet model in a 20ga, with 30" barrels, and at close to 8 lbs... ( heavier gun meant less recoil, better follow thru - and she thought better balance in her hands ...and she was only 5'6" and maybe 110 lbs soaking wet ...) ...and it made her husband think about what he thought a 20ga was ..or ought to be...
but before you drop $2,000 on a gun - make sure its exactly what you want ...but the Browning Super Light might work for you with 26" barrels - even though it wouldn't work for me...