Does every infinite-dimensional Banach space admit an "almost orthonormal" infinite set?

Question

Every infinite-dimensional Hilbert space admits an orthonormal basis. The basis has the three properties:

1. The elements are linearly independent

2. All elements have length $1$.

3. The pairs between any pair is $\sqrt 2$.

In a general Banach space define a set $S$ to be almost orthonormal to mean the following:

1. The elements of $S$ are linearly independent

2. There exists $E,\epsilon >0$ such that $E \ge \|s\| \ge \epsilon $ for all $s \in S$.

3. There exists $\delta >0$ such that $\|s_1-s_2\| \ge \delta$ for all $s_1,s_2 \in S$.

Does every Banach space contain an infinite almost orthonormal set?