Is the mode considered a resistant statistic?
By the definition of resistant given in our text (i.e., not changed much by outliers), I would think the answer would be "yes."
On the other hand, the definition of resistant given here (https://www.stat.berkeley.edu/~stark/SticiGui/Text/gloss.htm) makes me think that the answer would be "no." If one of the repeated data values was accidentally changed to something else, then one might not be able to recognize the mode as such.
As I commented, you are going to have to tell us how you find the mode, particularly for a sample from a continuous distribution where all the values are different
There are estimators for the mode which are sensitive to outliers, and others which are usually more robust such as the half sample mode, which is relatively easy to explain recursively (at least before ties are considered) with an algorithm like:
The half sample mode of $n$ sample observations is the half sample mode of those observations which lie in the narrowest interval which contains at least $n/2$ of the observations. When this reduces to $n=2$ observations, take the mean of these two.
It should be obvious that this particular estimator will be relatively resistant to extreme outliers which are not close to each other and indeed it will be more resistant even than the media in such cases. What is it less resistant to is spurious observations which are close to each other or close to genuine observations which are away from the mode.
You can read more about this and other robust estimators of the mode in a 2005 paper by David R. Bickel and Rudolf Fruehwirth, On a Fast, Robust Estimator of the Mode: Comparisons to Other Robust Estimators with Applications
