This may just be semantics, but are they awarded a new series, or a single down? I know it's just a single play, but when we say they are awarded a new series - which by common definition is more than one - it could be confusing for someone who comes across this.
Officially, they are awarded a series, if they are in regular scrimmage. But, because time in the period has expired, they will play until a down is played without an accepted penalty. Yeah, that will be 1st down, played over and over, if necessary, but that is a series. No different than if there was one second remaining in the period. They get a series, but, there is no way they’ll get past 1st down. Now, the untimed down may be a free kick, and not a down of a regular scrimmage series. But, if regular scrimmage, it is, officially, a series.