The only Buddhist monastery in the Taxila valley was a thriving centre of learning at the end of the third century AD.
The monastery attracted so many students and monks from around greater India that its administration built an annex to house the seekers of enlightenment coming to meditate there, archaeologists at Quaid-e-Azam University (QAU) have discovered.
Another notable finding was that the main compound of the monastery, located in present-day Badal Pur, is at least 300 years older than archaeologists previously estimated. The main compound, which consists of 55 “monk cells”, was excavated between 2005 and 2012.
http://tribune.com.pk/story/527525/myst ... la-valley/
This is especially noteworthy:
Khan said pieces of charcoal found at the monastery were analysed in the US at the University of Wisonsin using carbon dating — a reliable scientific method for determining the age of certain materials.
The university’s report states the monastery was built somewhere around the third century BC, making it one of the oldest monasteries in the Gandhara region.
If that's true, this would have been built in Aśoka's time or thereabouts.