Prove unbiasedness for $T(X_1,...,X_n) = X_1$ with $X_1 \sim Bin(1,p)$ and $X_i$ i.i.d.

Question

I have $X_1,...,X_n$ i.i.d. random variables with $X_i \sim Bin(1,p)$, given the estimator $T(X_1,...,X_n) = X_1$ I want to show that $T$ is a biased estimator.
If I try to I get this result: $E(X_1) = p$ since $Bin(1,p)$'s mean is $p$.
Since the random variables $X_i$ are identically distributed doesn't that mean that the estimator is unbiased since all of them have a mean of $p$?