The collective behavior of K0S and Λ/Λ¯¯¯¯ strange hadrons is studied by measuring the elliptic azimuthal anisotropy (v2) using the scalar-product and multiparticle correlation methods. Proton-lead (pPb) collisions at a nucleon-nucleon center-of-mass energy sNN−−−√ = 8.16 TeV and lead-lead (PbPb) collisions at sNN−−−√ = 5.02 TeV collected by the CMS experiment at the LHC are investigated. Nonflow effects in the pPb collisions are studied by using a subevent cumulant analysis and by excluding events where a jet with transverse momentum greater than 20 GeV is present. The strange hadron v2 values extracted in pPb collisions via the four- and six-particle correlation method are found to be nearly identical, suggesting the collective behavior. Comparisons of the pPb and PbPb results for both strange hadrons and charged particles illustrate how event-by-event flow fluctuations depend on the system size.