Yes, such examples do live. The following answer was given in "Metric spaces of non-positive curvature" by Bridson and Haefliger; thanks to [GGT][1] and [Moishe Kohan][2].

[![enter image description here][3]][3]


  [1]: https://mathoverflow.net/users/45170/ggt
  [2]: https://mathoverflow.net/users/39654/moishe-kohan
  [3]: https://i.sstatic.net/mjyGS.png