Unsure of base award

[EdMc]

Fed rules....No one on base. Batter hits hot smash to 3rd baseman who overthrows to 1st base. By the time the 1st baseman gets to the ball the batter has rounded 1st and going for 2nd. 1st baseman picks up ball and promptly falls completely into dead ball territory. What is the proper base award? Second or third base? Is the batter/runner now considered just a runner? Is it unintentional 'catch and carry'? We gave him second, using the 1st throw from infielder ruling. However, the throw didn't go out of bounds. The first baseman picked up the ball, THEN he and the ball went into DBT.
I have that sinking feeling we got it wrong. Help please.

[TomClarke]

you got it right for the wrong reason. runners get 1 base when a fielder falls into dead ball territory. his last occupied base was first, so he gets second.

[mr umpire]

Does the 1 base only apply to Fed or to all codes?