There is a Base64 quasi-fixed point:

    $ echo -n Vm0 | base64
    Vm0w

For reference, a program to generate the quasi-fixed point from scratch:

  #!/usr/bin/env python3
  import base64

  def len_common_prefix(a, b):
      assert len(a) < len(b)
      for i in range(len(a)):
          if a[i] != b[i]:
              return i
      return len(a)

  def calculate_quasi_fixed_point(start, length):
      while True:
          tmp = base64.b64encode(start)
          l = len_common_prefix(start, tmp)
          if l >= length:
              return tmp[:length]
          print(tmp[:l].decode('ascii'), tmp[l:].decode('ascii'), sep='\v')
          # Slicing beyond end of buffer will safely truncate in Python.
          start = tmp[:l*4//3+4] # TODO is this ideal?

  if __name__ == '__main__':
      final = calculate_quasi_fixed_point(b'\0', 80)
      print(final.decode('ascii'))

  Vm0wd2QyUXlVWGxWV0d4V1YwZDRWMVl3WkRSV01WbDNXa1JTVjAxV2JETlhhMUpUVmpBeFYySkVUbGho

Note that the suffix will grow in length with the input, making it less and less interesting.

(Because the output is necessarily 8/6 the size of the input, the suffix always adds 33% to the length.)

From the other direction, you'd call it a tail-eating unquine?