A quantum system interacts with its environment, if ever so slightly, no matter how much care isput into isolating it. As a consequence, quantum bits (qubits) undergo errors, putting dauntingly difficult constraints on the hardware suitable for quantum computation. New strategies are emerging to circumvent this problem by encoding a qubit non-locally across the phase space of a physicalsystem. Since most sources of decoherence are due to local fluctuations, the foundational promise isto exponentially suppress errors by increasing a measure of this non-locality. Prominent examples are topological qubits which delocalize quantum information over real space and where spatial extent measures non-locality. In this work, we encode a qubit in the field quadrature space of asuperconducting resonator endowed with a special mechanism that dissipates photons in pairs. This process pins down two computational states to separate locations in phase space. As we increasethis separation, we measure an exponential decrease of the bit-flip rate while only linearly increasing the phase-flip rate. Since bit-flips are continuously and autonomously corrected at the single qubit level, only phase-flips are left to be corrected via a one-dimensional quantum error correction code.This exponential scaling demonstrates that resonators with non-linear dissipation are promisingbuilding blocks for universal fault-tolerant quantum computation with drastically reduced hardware overhead.