Dare we bring up the "i" word?  ;-)

Voicemail is a network feature.  There have been various "answering
machine" programs for smartphones, that auto-answer if you don't, and
record the message.

I remember seeing such an app for Symbian phones years ago (back when I
used a Nokia 3650) but remember nothing about it except it was more money
than I was willing to pay.

However, these days you can use (the i-word again!)-style Visual
Voicemail with many smartphones.  Callwave.com offers a free V-V service.
You forward your phone to their number when "unavailable" (phone off,
busy, off-network, unanswered, etc.) and they record the message, then e-
mail it to you as a .wav file attachment.  On my Windows Mobile phone, I
receive them to my push e-mail account within a minute of them being
left, and I can tap them to play them, or download and archive them on my
PC.

So, to answer your question, any smartphone with push e-mail and a
media/.wav file player can store your voicemail on the phone (though not
by recording it directly like you asked.  So a server is still involved,
but the advantage is that if your phone is turned off or out-of-network,
you'll get the .wavs pushed as soon as it's back on.)