Is it possible to exercise an option for services provided under a BPA through a bilateral modification after the base period has expired, and if so, what authority allows for it?
No, because FAR 13.303-7 defines your BPA as complete you would not be able to exercise the option on this closed BPA. I would also say that an option, as the term is defined within FAR 2.101, shouldn't really be used with BPA's in the first place. The FAR envisions Agreements that are longer than a year because it talks about annual reviews and the need to amend the agreements. In the future, you can establish a longer term agreement and then rely on the FAR's guidance for a 30 day written notice to dissolve the BPA by either party.
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In your current scenario, if only a couple of days have elapsed, you could consider reestablishing the Agreement with the same contractor. This seems inappropriate based on competition laws/regulations, but FAR 13.501 outlines a less restrictive justification write-up for non-competitive actions when using Simplified Acquisition Procedures. Your legal support (and you should engage them) will likely want to consider several factors to include the length of time the original BPA has been expired, how the BPA was originally captured in the Contract Action Report (CAR), and the dollar ceiling on the agreement.