  We construct a motivic homotopy theory for rigid analytic varieties with the
rigid analytic affine line $\mathbb{A} ^1_\mathrm{rig}$ as an interval object.
This motivic homotopy theory is inspired from, but not equal to, Ayoub's
motivic homotopy theory for rigid analytic varieties. Working in the so
constructed homotopy theory, we prove that a homotopy classification of vector
bundles of rank $n$ over rigid analytic quasi-Stein spaces follows from
$\mathbb{A}^1_\mathrm{rig}$-homotopy invariance of vector bundles. This
$\mathbb{A}^1_\mathrm{rig}$-homotopy invariance is equivalent to a rigid
analytic version of Lindel's solution to the Bass--Quillen conjecture.
Moreover, we establish a homotopy classification of line bundles over rigid
analytic quasi-Stein spaces. In fact, line bundles are classified by infinite
projective space.
