Is there any example of this velcro-like space? Looking for a metric space $(X,d)$ such that:

(A)-it has a metric tangent space $(T_{x}X, d^x)$ at any point $x \in X$,

(B)-the set $X$ is topologically a surface,

(C)-for any $x \in X$ the metric tangent space $(T_{x}X, d^x)$ is not isometric with the unit ball in a normed group.

Normed group: a metric space $(G,d)$ such that $G$ is a group and $d$ is a left-invariant distance. Let $e$ be the neutral element of $G$. Then for any $x \in G$ the norm of $x$ is $\| x \| = d(e,x)$. The unit ball is the closed ball of center $e$ and radius $1$.

Condition (B) may be relaxed to "X admits a differential structure", although I think that it might be misleading. The question has a purely metric character.

Is there any example of this velcro-like space? Looking for a LOCALLY COMPACT COMPLETE metric space $(X,d)$ such that:

(A)-it has a metric tangent space $(T_{x}X, d^x)$ at any point $x \in X$,

(B)-the set $X$ (with the topology given by $d$) is topologically a surface,

(C)-for any $x \in X$ the metric tangent space $(T_{x}X, d^x)$ is not isometric with the unit ball in a normed group.

Normed group: a metric space $(G,d)$ such that $G$ is a group and $d$ is a left-invariant distance. Let $e$ be the neutral element of $G$. Then for any $x \in G$ the norm of $x$ is $\| x \| = d(e,x)$. The unit ball is the closed ball of center $e$ and radius $1$.

Condition (B) may be relaxed to "X admits a differential structure", although I think that it might be misleading. The question has a purely metric character.