Using bootstrap to obtain sampling distribution of 1st-percentile

I have a sample (of size 250) from a population. I do not know the distribution of the population.

The main question: I want a point estimate of the 1st-percentile of the population, and then I want a 95% confidence interval around my point estimate.

My point estimate will be the sample 1st-percentile. I denote it $x$.

After that, I try to build the confidence interval around the point estimate. I wonder if it makes sense to use bootstrap here. I am very inexperienced with bootstrap, so pardon if I fail to use the appropriate terminology etc.

Here is how I tried to do it. I draw 1000 random samples with replacement from my original sample. I obtain the 1st-percentile from each of them. Thus I have 1000 points - "the 1st-percentiles". I look at the empirical distribution of these 1000 points. I denote the mean of it $x_{mean}$. I denote a "bias" as follows: $\text{bias}=x_{mean}-x$. I take the 2.5th-percentile and 97.5th percentile of the 1000 points to obtain the lower and the higher end of what I call a 95% confidence interval around the 1st-percentile of the original sample. I denote these points $x_{0.025}$ and $x_{0.975}$.

The last remaining step is to adapt this confidence interval to be around the 1st-percentile of the population rather than around the 1st-percentile of the original sample. Thus I take $x-\text{bias}-(x_{mean}-x_{0.025})$ as the lower end and $x-\text{bias}+(x_{0.975}-x_{mean})$ as the upper end of the 95% confidence interval around the point estimate of the population's 1st-percentile. This last interval is what I was seeking for.

A crucial point, in my opinion, is whether it makes sense to use bootstrap for 1st-percentile which is rather close to the tail of the unknown underlying distribution of the population. I suspect it might be problematic; think about using bootstrap for building a confidence interval around a minimum (or a maximum).

But perhaps this approach is flawed? Please let me know.

EDIT:

Having thought about the problem a little more, I see that my solution implies the following: the empirical 1st percentile of the original sample may be a biased estimator of the 1st percentile of the population. And if so, the point estimate should be bias-adjusted: $x-\text{bias}$. Otherwise the bias-adjusted confidence interval would not be compatible with the bias-unadjusted point estimate. I need to adjust either both the point estimate and the confidence interval or none of them.

If, on the other hand, I did not allow for the estimate to be biased, I would not have to do the bias adjustment. That is, I would take $x$ as the point estimate and $x-(x_{mean}-x_{0.025})$ as the lower end and $x+(x_{0.975}-x_{mean})$ as the upper end of the 95% confidence interval. I am not sure whether this interval makes sense...

So does it make any sense to assume that the sample 1st percentile is a biased estimate of the population 1st percentile? And if not, is my alternative solution correct?

Bootstrap inference for the extremes of a distribution is generally dubious. When bootstrapping n-out-of-n the minimum or maximum in the sample of size $n$, you have $1 - (1-1/n)^n \sim 1 - {\rm exp}(-1) = 63.2\%$ chance that you will reproduce your sample extreme observation, and likewise approximately ${\rm exp}(-1) - {\rm exp}(-2)=23.3\%$ chance to reproduce your second extreme observation, and so on. You get a deterministic distribution that has little to do with the shape of the underlying distribution at the tail. Moreover, the bootstrap cannot give you anything below your sample minimum, even when the distribution has the support below this value (as would be the case with most continuous distributions like say normal).
The solutions are complicated and rely on the combinations of asymptotics from extreme value theory and subsampling fewer than n observations (actually, way fewer, the rate should converge to zero as $n\to\infty$).