Is there a Nash equilibrium in Bertrand model with cost advantages?

No, if prices are continuous, no equilibrium exists. This is usually called the "open set problem". As you write in your explanation the player with the cost advantage wants to undercut the other player by a small amount, say $\epsilon$ to $c_2-\epsilon$. However, for all $\epsilon>0$, there is a better strategy. For example, setting $p_1=c_2-\epsilon/2$. With a similar argument you can rule out mixed strategies.

There are usually two approaches to fix this. First, assume that prices have to be discrete. In this case, the equilibrium is that the weak player chooses the lowest price on the price grid that is weakly above $c_2$ and the strong player chooses the next lower price on the price grid. Second, assume that ties are broken in favor of the stronger player. That is, if both players set the same price, the strong player gets all the demand. This can make sense if you interpret the Bertrand game as a first-price auction.