Speaking to Famitsu during Tokyo Game Show 2018, Sony has confirmed that the PlayStation Vita will cease production at least in Japan sometime in 2019. Presumably, it would phase out at the end of Sony's Fiscal Year which closes on March 31, 2019.
This follows on the heels of the news a few months ago (okay, very long heels) that physical PlayStation Vita games will end production on February 15, 2019. Chances are, this news will affect the rest of the world considering Sony is obviously a Japanese company and these things tend to happen last there, especially in a territory that fell far more in love with the handheld than many other territories in the world.
Finally, in that same interview, Sony Interactive Entertainment senior vice president Hiroyuki Oda has said they have no plans to work on a successor for the PlayStation Vita. Honestly, with how things went with the PS Vita, it's hard to say whether that's a good or bad thing, but competition is always a healthy thing.