Sony has produced some cool, stylish portables and has done well overseeing the systems' worthwhile games.

I think the most they would do next is test a new handheld in Japan, and maybe Australia and go from there.

No, I don't want Sony wasting more money in the handheld market

if they do i fear for the company. the market is gone and has moved to mobile ios and android. Nintendo will not last much longer either and they are kings of handheld gaming.

Don't think so...
The Handheld market is dying,plus they can't risk to have another failure like the Vita.

For me Sony made three crucial mistakes with Vita:

- Overengineeredthe device

Vita is a full generational jump from PSP.  The jump in perfomance is comparable to the one from PS2 to PS3.    

This made Sony release an expansive platform in the market in a time when Nintendo had already tasted with the 3DS first months in Japan that there is no market for a $250 handheld anymore.  Nintendo had the wisdom to cut the price of the console fast while Sony was dumb enough not to take notes from this and released the platform at a high price anyway.   

I really believe that Sony shouldn't have designed the Vita to be so much more powerful than the PSP.   Vita's main competitor, the 3DS is in many ways an  enhanced PSP and consumers are fine with it.     A redesigned PSP with Two analog sticks + Camera + improved screen could have been way more attractive.

-  Too many console like games for a Handheld.

Games like Uncharted, Borderlands 2, LBP and others are cool on HOME CONSOLES but not as attractive on a handheld platform.  With few exceptions like Gravity Rush, Soul Sacrifice and Tearaway.  Vita sometimes looks like  a platform to play games that can be enjoyed in a better way on the PS3.

- Losing Monster Hunter to Nintendo.

In my opinion this was the final nail in Vita's coffin.    In many ways this reminds me of Final Fantasy VII situation during the N64/PS1 days.

I still can't believe Sony just let Nintendo secure exclusivity of this game.  Specially considering how important this franchise was to the PSP.

 MH was one of the games that defined and made the PSP succcessful in Japan.

Damn, poll is 0 - 15 at the time I voted...
And those 15 are correct. Sony is done with handheld, same as the world is done with handhelds.

They will probably shift their focus toward mobile services (such as PS Now).

I don't want them to and I don't think they will.

4 trolls have voted "yes" ^^ and why do I call them trolls? Because none have offered an explanation in the comments, that's why I assume they're trolls. So don't ban me.

Is there really any argument that would support Sony going with another handheld?

The only thing I can think of is if they find a way to make some sort of flip phone like the PSP GO was and add a touch screen. Even then I am not sure that it would be a wise idea.

Nope. Streaming to mobile devices will be the thing going forward.(Except for Nintendo)