What does it mean having 1 as best k parameter in K-NN? I'm working with a large dataset (761 rows and about 57k-60k features) and after doing a feature selection to select the best 10 features I'm using different ML algorithms to classify some cases. In the case of the K-NN I performed a hyperparameter tunning using GridSearchCV from sklearn and after many tests this returns 1 as the best value for k.
I tried with the RandomizedSearch too and increasing n_iter parameter makes it to eventually return 1 aswell as the best value for k.
Does this mean something? It seems strange to me having this value as the best for k. Could it be overfitting maybe?
Thanks!

There is nothing "bad" about $k=1$. It's a hyperparameter to tune, so different values would work for different problems. If you did your hyperparameter tuning correctly, i.e. there are no data leaks in cross-validation, then it's not overfitting because the parameter was validated on external data. If you worry about overfitting, use the held-out test set to verify the final solution, if it is overfitted then the model would perform poorly on the test set.
If $k=1$ is the best, this may simply mean that your data can be easily explained by a fixed set of "typical cases" that you just need to memorize for making predictions. With higher values of $k$ you would average over multiple examples to reduce noise, maybe here it unnecessarily "blurs" the predictions and noise is not an issue. Of course, those are just guesses, you would need to conduct an in-depth exploratory analysis of the predictions to answer the "why" question.
