CJam, 43 39 35 bytes

"<~"l4/{:L4Ue]256b85b'!f+}/L,)<"~>"

Try it online in the [CJam interpreter].

How it works

"<~"      e# Push that string.
l4/       e# Read a line from STDIN and split it into chunks of length 4.
{         e# For each chunk:
  :L      e#   Save it in L.
  4Ue]    e#   Right-pad it with 0's to a length of 4.
  256b85b e#   Convert from base 256 to base 85.
  '!f+    e#   Add '!' to each base-85 digit.
}/        e#
L,)       e# Push the length of the last unpadded chunk, plus 1.
<         e# Keep that many chars of the last encoded chunk.
"~>"      e# Push that string.

We don't have to deal with z or pad the encoded chunks to length 5, since the input will contain only printable ASCII characters.

[CJam interpreter]: http://cjam.aditsu.net/#code=%22%3C~%22l4%2F%7B%3AL4Ue%5D256b85b'!f%2B%7D%2FL%2C)%3C%22~%3E%22&input=Programming%20Puzzles

CJam, 43 39 35 bytes

"<~"q4/{:N4Ue]256b85b'!f+}/N,)<"~>"

Try it online in the [CJam interpreter].

How it works

"<~"      e# Push that string.
q4/       e# Read all input from STDIN and split it into chunks of length 4.
{         e# For each chunk:
  :N      e#   Save it in N.
  4Ue]    e#   Right-pad it with 0's to a length of 4.
  256b85b e#   Convert from base 256 to base 85.
  '!f+    e#   Add '!' to each base-85 digit.
}/        e#
N,)       e# Push the length of the last unpadded chunk, plus 1.
<         e# Keep that many chars of the last encoded chunk.
"~>"      e# Push that string.

If the input was empty, N,) will apply to the string "<~". Since N initially holds a single character, the output will be correct.

We don't have to deal with z or pad the encoded chunks to length 5, since the input will contain only printable ASCII characters.

[CJam interpreter]: http://cjam.aditsu.net/#code=%22%3C~%22q4%2F%7B%3AN4Ue%5D256b85b'!f%2B%7D%2FN%2C)%3C%22~%3E%22&input=Programming%20Puzzles

CJam, 43 39 bytes

"<~"l4/{:L4Ue]256b85b5Ue['!f+}/L,)<"~>"

Try it online in the [CJam interpreter].

[CJam interpreter]: http://cjam.aditsu.net/#code=%22%3C~%22l4%2F%7B%3AL4Ue%5D256b85b5Ue%5B'!f%2B%7D%2FL%2C)%3C%22~%3E%22&input=Programming%20Puzzles

CJam, 43 39 35 bytes

"<~"l4/{:L4Ue]256b85b'!f+}/L,)<"~>"

Try it online in the [CJam interpreter].

How it works

"<~"      e# Push that string.
l4/       e# Read a line from STDIN and split it into chunks of length 4.
{         e# For each chunk:
  :L      e#   Save it in L.
  4Ue]    e#   Right-pad it with 0's to a length of 4.
  256b85b e#   Convert from base 256 to base 85.
  '!f+    e#   Add '!' to each base-85 digit.
}/        e#
L,)       e# Push the length of the last unpadded chunk, plus 1.
<         e# Keep that many chars of the last encoded chunk.
"~>"      e# Push that string.

We don't have to deal with z or pad the encoded chunks to length 5, since the input will contain only printable ASCII characters.

[CJam interpreter]: http://cjam.aditsu.net/#code=%22%3C~%22l4%2F%7B%3AL4Ue%5D256b85b'!f%2B%7D%2FL%2C)%3C%22~%3E%22&input=Programming%20Puzzles

CJam, 43 bytes

"<~"q4/{:L4Ue]256b_85b5Ue['!f+'z?}/L,)<"~>"

Try it online in the [CJam interpreter].

[CJam interpreter]: http://cjam.aditsu.net/#code=%22%3C~%22q4%2F%7B%3AL4Ue%5D256b_85b5Ue%5B'!f%2B'z%3F%7D%2FL%2C)%3C%22~%3E%22&input=Programming%20Puzzles

CJam, 43 39 bytes

"<~"l4/{:L4Ue]256b85b5Ue['!f+}/L,)<"~>"

Try it online in the [CJam interpreter].

[CJam interpreter]: http://cjam.aditsu.net/#code=%22%3C~%22l4%2F%7B%3AL4Ue%5D256b85b5Ue%5B'!f%2B%7D%2FL%2C)%3C%22~%3E%22&input=Programming%20Puzzles