In this article, an extension of the L-1 adaptive control design is introduced for a class of non-affine Multi-Input Multi-Output nonlinear systems with unknown dynamics and unmeasured states. The system dynamics is represented in the normal form with the bounded-input-bounded-output internal dynamics. At first, a stable virtual reference counterpart is constructed. Thereafter, a piece-wise continuous adaptive law is introduced to the actual system along with a low-pass filtered control signal that allows for achieving arbitrarily close tracking of the input and the output signals of the reference system. Rigorous mathematical proof is provided, and the theoretical results are verified with the simulation.