  We develop a new twistorial field formulation of a massless infinite spin
particle. Unlike our previous approach arXiv:1805.09706, the quantization of
such a world-line infinite spin particle model is carried without any gauge
fixing. As a result, we construct a twistorial infinite spin field and derive
its helicity decomposition. Using the field twistor transform, we construct the
space-time infinite (continuous) spin field, which depends on the coordinate
four-vector and additional commuting Weyl spinor. The equations of motion for
infinite spin fields in the cases of integer and half-integer helicities are
derived. We show that the infinite integer-spin field and infinite
half-integer-spin field form the $\mathcal{N}{=}\,1$ infinite spin
supermultiplet. The corresponding supersymmetry transformations are formulated
and their on-shell algebra is derived. As a result, we find the field
realization of the infinite spin $\mathcal{N}{=}\,1$ supersymmetry.
