The non-abelian Higgs (NAH) theory is studied in a strong magnetic field. For simplicity, we study the SU(2) NAH theory with the Higgs triplet in a constant strong magnetic field $\vec B$, where the lowest-Landau-level (LLL) approximation can be used. Without magnetic fields, charged vector fields $A_\mu^\pm$ have a large mass $M$ due to Higgs condensation, while the photon field $A_\mu$ remains to be massless. In a strong constant magnetic field near and below the critical value $eB_c \equiv M^2$, the charged vector fields $A_\mu^\pm$ behave as 1+1-dimensional quasi-massless fields, and give a strong correlation along the magnetic-field direction between off-diagonal charges coupled with $A_\mu^\pm$. This may lead a new type of confinement caused by charged vector fields $A_\mu^\pm$.