Can we estimate the mean of an asymmetric distribution in an unbiased and robust manner?

This is not an unbiased estimate, but it is consistent (you can let the bias approach to zero as the sample size grows).

You can take a trimmed sample (remove the highest and lowest values) and use the mean of the trimmed sample as the estimate.

In the case of a know distribution then you might use an appropriate scaling to make the estimate less biased (or not biased at all), or otherwise the bias will just decrease when you take smaller samples.

As already said by whuber, one way to answer your question is to de-biase your estimator. If the robust estimator is biased, maybe you can subtract the theoretical bias (according to a parametric model), there are some work that try to do that or to subtract an approximation of the bias (I don't remember a ref but I could search for it if you are interested). For instance, think about the empirical median in an exponential model. We can compute its expectation and then substract this expectation, if you want I can make the computations this is rather simple ... this becomes more difficult if the estimator is more complicated than the median and this works only in parametric models.

A maybe less ambitious question is whether we can construct a consistent robust estimator. This we can do but we have to be careful of what we call robust.

If your definition of robust is having a non-zero asymptotic breakdown point, then already we can prove that this is impossible. Suppose that your estimator is called $T_n$ and it converges to $mathbb{E}[X]$. $T_n$ has a non-zero breakdown point which means that there can be a portion $varepsilon>0$ of the data arbitrarily bad and nonetheless $T_n$ will not be arbitrarily large. But this can't be because at the limit, if a portion of the data is an outlier, this translates: with probability $1-varepsilon$, $X$ is sampled from the target distribution $P$ and with probability $varepsilon$ $X$ is arbitrary, but this makes $mathbb{E}[X]$ arbitrary also (if you want me to put it formally, I can) which is in contradiction with the non-asymptotic breakdown point of $T_n$.

Finally, to conclude on this, we can take the non-asymptotic point of view. Saying that we don't care about the asymptotic breakdown point, what is important is either a the non-asymptotic breakdown point (something like a breakdown point of $1/sqrt{n}$. Or to be efficient on heavy-tailed data.

In this case, there are estimators that are robust and consistent estimators of $mathbb{E}[X]$. For instance, we can use Huber's estimator with a parameter that goes to infinity or we can use the median-of-means estimator with a number of blocks that tends to infinity. References for this line of thought are "Challenging the empirical mean and empirical variance: A deviation study" by Olivier Catoni or "Sub-Gaussian mean estimators" by Devroye et al (these ref are in the theoretical community, they may be complicated if you are not familiar with empirical processes and concentration inequalities).