Λ warm dark matter (ΛWDM), realized by collisionless particles of 1–3 keV, has been proposed as an alternative scenario to Λ-Cold-Dark Matter (ΛCDM) for the dwarf galaxy scale discrepancies. We present an approach to test the viability of such WDM models using star-formation histories (SFHs) of the dwarf spheroidal galaxies (dSphs) in the Local Group. We compare their high-time-resolution SFHs with the collapse redshift of their dark halos in CDM and WDM. Collapse redshift is inferred after determining the subhalo infall mass. This is based on the dwarf current mass inferred from stellar kinematics, combined with cosmological simulation results on subhalo evolution. WDM subhalos close to the filtering mass scale, forming significantly later than CDM, are the most difficult to reconcile with early truncation of star formation (z ≥ 3). The ultra-faint dwarfs (UFDs) provide the most stringent constraints. Using six UFDs and eight classical dSphs, we show that a 1 keV particle is strongly disfavored, consistently with other reported methods. Excluding other models is only hinted for a few UFDs. Other UFDs for which the lack of robust constraints on halo mass prevents us from carrying out our analysis rigorously, show a very early onset of star formation that will strengthen the constraints delivered by our method in the future. We discuss the various caveats, notably the low number of dwarfs with accurately determined SFHs and the uncertainties when determining the subhalo infall mass, most notably the baryonic physics. Our preliminary analysis may serve as a pathfinder for future investigations that will combine accurate SFHs for local dwarfs with direct analysis of WDM simulations with baryons.