When is there time to program?

When is there time to program?

In the old days, programmers were kept behind the curtain coding away..
isolated from the whirlwind of operations and support. But in today's
agile world where collaboration and communication are staples, programers
are drawn into the mire.
So, when the heck are we supposed to actually code?
Even if we designate chunks of time to "leave coders alone", they'll
emerge to 57 emails waiting for them, many that needed responses hours
ago.
To say "evenings and weekends" kind of stinks but it's what I for one
require as normal work hours are too chaotic to get any isolated,
deep-thinking coding done.
In large organizations at least support can be segmented to a different
group. But ours is a smaller shop (like so many others) where we all have
to wear many hats. But we're getting less coding throughput given all of
the other stuff.
Opinions/advice/suggestions?
thanks