Yes, it is both, an Ito process and a semimartingale, if you are ready to enlarge the filtration. We have

$$Y_t = \int_0^t I\bigl( s \in [0,t-1) \bigr) \, dB_s.$$

Note in particular that for $t\leq 1$ the indicator yields $0$ trivially. However, the integrand is not measurable in the Brownian filtration. However, if we enlarge the filtration to encompass also all indicator functions, we get a martingale with respect to this filtration which is an "Ito proceess" (or maybe better called "Skorohod process" as this is a Skorohod integral).