Your raster (2015LSTDay.tif, band 5)

You can use rasterio with pandas:
import rasterio as rio
import pandas as pd
with rio.open('2015LSTDay.tif') as dataset:
val = dataset.read(5) # band 5
no_data=dataset1.nodata
data = [(dataset1.xy(x,y)[0],dataset1.xy(x,y)[1],val[x,y]) for x,y in np.ndindex(val.shape) if val[x,y] != no_data]
lon = [i[0] for i in data]
lat = [i[1] for i in data]
d = [i[2] for i in data]
res = pd.DataFrame({"long":lon,'lat':lat,"data":v})
res.head()
long lat val
0 12.307060 42.0571 15631.0
1 12.315384 42.0571 15647.0
2 12.323708 42.0571 15624.0
3 12.332033 42.0571 15624.0
4 12.340357 42.0571 15590.0
Result
import matplotlib.pyplot as plt
from rasterio.plot import show
fig, ax = plt.subplots(figsize=(8, 8))
show(dataset.read(5), transform=dataset.transform,ax=ax)
ax.plot(res.x,res.y,'ro', markersize=3)

You can use rasterio with GeoPandas:
import geopandas as gpd
from shapely.geometry import Point
with rio.open('2015LSTDay.tif') as dataset:
val = dataset.read(5) # band 5
no_data=dataset.nodata
geometry = [Point(dataset.xy(x,y)[0],dataset.xy(x,y)[1]) for x,y in np.ndindex(val.shape) if val[x,y] != no_data]
v = [val[x,y] for x,y in np.ndindex(val.shape) if val[x,y] != no_data]
df = gpd.GeoDataFrame({'geometry':geometry,'data':v})
df.crs = dataset.crs
df.head()
geometry data
0 POINT (12.30706 42.05710) 15631.0
1 POINT (12.31538 42.05710) 15647.0
2 POINT (12.32371 42.05710) 15624.0
3 POINT (12.33203 42.05710) 15624.0
4 POINT (12.34036 42.05710) 15590.0
Export to shapefile
df.to_file("points.shap")
You can also use rioxarray as suggested by snowman2:
import rioxarray
rds = rioxarray.open_rasterio("2015LSTDay.tif")
rds = rds.squeeze().drop("spatial_ref").drop("band")
rds.name = "data"
res = rds.to_dataframe().reset_index()
res.head(2)
band y x data
0 0 42.0571 12.307060 15228.0
1 0 42.0571 12.315384 15246.0
Band 5 only
gr = res.groupby(res.band)
gr.get_group('5').head()
band y x data
11550 5 42.0571 12.307060 15652.0
11551 5 42.0571 12.315384 15671.5
11552 5 42.0571 12.323708 15702.5
11553 5 42.0571 12.332033 15702.5
11554 5 42.0571 12.340357 15642.5