  We measure the star formation histories (SFH) of a sample of low-mass
galaxies with $M_\ast<10^9M_\odot$ from the SDSS-IV MaNGA survey. The large
number of IFU spectra for each galaxy are either combined to reach a high
signal to noise ratio or used to investigate spatial variations. We use
Bayesian inferences based on full spectrum fitting. Our analysis based on
Bayesian evidence ratio indicates a strong preferencefor a model that allows
the presence of an old stellar population, and that an improper model for the
SFH can significantly underestimate the old population in these galaxies. The
addition of NIR photometry to the constraining data can further distinguish
between different SFH model families and significantly tighten the constraints
on the mass fraction in the old population. On average more than half of the
stellar mass in present-day low-mass galaxies formed at least 8 Gyrs ago, while
about 30\% within the past 4 Gyrs. Satellite galaxies on average have formed
their stellar mass earlier than central galaxies. The radial dependence of the
SFH is quite weak. Our results suggest that most of the low-mass galaxies have
an early episode of active star formation that produces a large fraction of
their present stellar mass.
