  Continuing with the study of Approximately ultrahomogeneous and Fra\"iss\'e
Banach spaces introduced by V. Ferenczi, J. L\'opez-Abad, B. Mbombo and S.
Todorcevic, we define formally weaker and in some aspects more natural
properties of Banach spaces which we call Almost ultrahomogeneity and the
Almost Fra\"iss\'e Property. We obtain relations between these different
homogeneity properties of a space $E$ and relate them to certain pseudometrics
on the class $\mathrm{Age}(E)$ of finite dimensional subspaces of $E$. We prove
that ultrapowers of an almost Fra\"iss\'e Banach space are ultrahomogeneous. We
also study two properties called finitely isometrically extensible and almost
finitely isometrically extensible, respectively, and prove that approximately
ultrahomogeneous Banach spaces are finitely isometrically extensible.

Functional Analysis