We implement the proof of principle for the quantum walk of one ion in a linear ion trap. With a single-step fidelity exceeding 0.99, we perform three steps of an asymmetric walk on the line. We clearly reveal the differences to its classical counterpart if we allow the walker or ion to take all classical paths simultaneously. Quantum interferences enforce asymmetric, nonclassical distributions in the highly entangled degrees of freedom (of coin and position states). We theoretically study and experimentally observe the limitation in the number of steps of our approach that is imposed by motional squeezing. We propose an altered protocol based on methods of impulsive steps to overcome these restrictions, allowing to scale the quantum walk to many, in principal to several hundreds of steps.