  We present a new algorithm for precision radial velocity (pRV) measurements,
a line-by-line (LBL) approach designed to handle outlying spectral information
in a simple but efficient manner. The effectiveness of the LBL method is
demonstrated on two datasets, one obtained with SPIRou on Barnard's star, and
the other with HARPS on Proxima Centauri. In the near-infrared, the LBL
provides a framework for m/s-level accuracy in pRV measurements despite the
challenges associated with telluric absorption and sky emission lines. We
confirm with SPIRou measurements spanning 2.7 years that the candidate
super-Earth on a 233-day orbit around Barnard's star is an artifact due to a
combination of time-sampling and activity. The LBL analysis of the Proxima
Centauri HARPS post-upgrade data alone easily recovers the Proxima b signal and
also provides a 2-sigma detection of the recently confirmed 5-day Proxima d
planet, but argues against the presence of the candidate Proxima c with a
period of 1900 days. We provide evidence that the Proxima c signal is
associated with small, unaccounted systematic effects affecting the HARPS-TERRA
template matching RV extraction method for long-period signals. Finally, the
LBL framework provides a very effective activity indicator, akin to the full
width at half maximum derived from the cross-correlation function, from which
we infer a rotation period of $92.1^{+4.2}_{-3.5}$ days for Proxima.
