Gauss-Markov and Asymptotic Properties

Is it true that Gauss-Markov assumptions (i.e. linearity, full rank, strict exogeneity, and $$\sigma^2 I$$) can imply "consistency" and "asymptotic normality" of the OLS estimator?

Or do we also need to make assumptions about the distribution of $$X$$ and $$e$$?

And can we say that Gauss-Markov assumptions are stronger (or weaker) than asymptotic assumptions?

Consider the indicator variable model $$Y_i = \beta_1 I_i(i = 1) + \beta_2 I_i(i \neq 1) + \epsilon_i$$, $$i = 1, \dots , n$$, where the $$\epsilon_i$$ are i.i.d. mean zero, not normal, but with common finite variance. Assume $$\beta_1 = \beta_2 = 0$$. This model satisfies the G-M conditions.
(The model is used in "event studies" in finance, where in the above model, $$i=1$$ indicates a possible "event" under study.)
Here, $$\hat \beta_1 =\epsilon_1$$ for all $$n$$. But $$\epsilon_1$$ is not zero (for any $$n$$, regardless of whether $$n$$ is "large,") and it is not normally distributed.