We investigate a lossy source compression problem in which both the encoder and decoder are equipped with a pre-trained sequence predictor. We propose an online lossy compression scheme that, under a 0-1 loss distortion function, ensures a deterministic, per-sequence upper bound on the distortion (outage) level for any time instant. The outage guarantees apply irrespective of any assumption on the distribution of the sequences to be encoded or on the quality of the predictor at the encoder and decoder. The proposed method, referred to as online conformal compression (OCC), is built upon online conformal prediction--a novel method for constructing confidence intervals for arbitrary predictors. Numerical results show that OCC achieves a compression rate comparable to that of an idealized scheme in which the encoder, with hindsight, selects the optimal subset of symbols to describe to the decoder, while satisfying the overall outage constraint.