I know the scenario is common in Phase-I clinical trial. There are other practical situations too where sample size is small but the subjects are measured repeatedly. Can anyone kindly share any thought?
Here are some excellent sources of data sets that are freely available. Good luck with your search!
1. Data and Story Library (DASL): https://dasl.datadescription.com/
DASL is a resource maintained at Carnegie-Mellon University which has a variety of data sets, some oriented toward a specific data analysis method, others could be handled in a variety of ways. Though the data sets are numerous, they tend to be small, that is, based on a relatively small number of cases. The data sets also tend to be somewhat dated. That doesn't decrease their utility, however. Perhaps the biggest deficiency is that there is often little information about the actual source of the data, explanation of how the data were collected, and, in a few instances, the choice of how the data are quantified. Some of the data may be fictitious--drawn from text examples rather than real data.
2. Australasian Data and Story Library (OzDASL): http://www.statsci.org/data/index.html
OzDASL is similar to DASL, though the data sets tend to be somewhat better documented.
3. National Center for Educational Statistics (NCES): https://nces.ed.gov/
NCES is the home for all of the data collected and summarized in the "Nation's Report Card," which is based on the National Assessment of Educational Progress (NAEP) test series. There is much you can find here; some of which is in summary form, and some of which can be obtained in raw form.
4. Inter-university Consortium for Political and Social Research (ICPSR): https://www.icpsr.umich.edu/icpsrweb/
ICPSR is a treasure trove of data sets, many of which are from very large studies, sometimes done longitudinally over many years. Most of the data sets come from survey-type studies, but you should have no problem finding something here.
5. StatSci.org list of data sets: http://www.statsci.org/datasets.html
Here's a large list of links to data sets, some of which have been listed above (like DASL), some from textbooks, some from elsewhere. If you can't find something from this site, you're just not trying!
6. HUD USER data sets: https://www.huduser.gov/portal/datasets/pdrdatas.html
From the department of Housing and Urban Development comes a variety of electronically-distributed data sets that can be accessed. These are data sets that have been obtained from the Office of Policy Development and Research (PD&R) at HUD.
7. Journal of Statistics Education Data Sets: http://jse.amstat.org/jse_data_archive.htm
A variety of data sets, usually larger than those in DASL, which are submitted by authors to accompany their articles in JSE. Each data set typically has a corresponding file with a description, though you can always consult the article.
8. Raw data from personality tests: https://openpsychometrics.org/_rawdata/
Just what the title implies. All files are comma separated values ("csv"), which can easily be read into most statistics packages.
9. Dr. Wuench's data sets for stat classes: http://core.ecu.edu/psyc/wuenschk/spss/spss-Data.htm
A variety of data sets, most of small to moderate size.
10. Data sets from Julie Pallant's text: http://spss.allenandunwin.com.s3-website-ap-southeast-2.amazonaws.com/data_files.html