In 2012, Voyager 1 became the first in situ probe of the very local interstellar medium1. The Voyager 1 Plasma Wave System has given point estimates of the plasma density spanning about 30 au of interstellar space, revealing a large-scale density gradient2,3 and turbulence4 outside the heliopause. Previous studies of the plasma density relied on the detection of discrete plasma oscillation events triggered ahead of shocks propagating outwards from the Sun and used to infer the plasma frequency and hence density5,6. We present the detection of a class of very weak, narrowband plasma wave emission in the Voyager 1 data that persists from 2017 onwards and enables the first steadily sampled measurement of the interstellar plasma density over about 10 au with an average sampling distance of 0.03 au. We find au-scale density fluctuations that trace interstellar turbulence between episodes of previously detected plasma oscillations. Possible mechanisms for the narrowband emission include thermally excited plasma oscillations and quasi-thermal noise, and could be clarified by new findings from Voyager or a future interstellar mission. The emission's persistence suggests that Voyager 1 may be able to continue tracking the interstellar plasma density in the absence of shock-generated plasma oscillation events.