Basemap is a great library for mapping faster than other python options, but there are some usual things I couldn't find how to do. Clipping a raster using a shape is one of them. Here's how do I do it

The output 
As usual, all the code can be found at
GitHub
Getting some data
The shapefile will be the border of Andorra, taken from
Natural Earth
The result is a little poor because the resolution is low, but works well for the example.
The script
from mpl_toolkits.basemap import Basemap
from matplotlib.path import Path
from matplotlib.patches import PathPatch
import matplotlib.pyplot as plt
from osgeo import gdal
import numpy
import shapefile
fig = plt.figure()
ax = fig.add_subplot(111)
sf = shapefile.Reader("ne_10m_admin_0_countries")
for shape_rec in sf.shapeRecords():
if shape_rec.record[3] == 'Andorra':
vertices = []
codes = []
pts = shape_rec.shape.points
prt = list(shape_rec.shape.parts) + [len(pts)]
for i in range(len(prt)  1):
for j in range(prt[i], prt[i+1]):
vertices.append((pts[j][0], pts[j][1]))
codes += [Path.MOVETO]
codes += [Path.LINETO] * (prt[i+1]  prt[i] 2)
codes += [Path.CLOSEPOLY]
clip = Path(vertices, codes)
clip = PathPatch(clip, transform=ax.transData)
m = Basemap(llcrnrlon=1.4,
llcrnrlat=42.4,
urcrnrlon=1.77,
urcrnrlat=42.7,
resolution = None,
projection = 'cyl')
ds = gdal.Open('srtm_37_04.tif')
data = ds.ReadAsArray()
gt = ds.GetGeoTransform()
x = numpy.linspace(gt[0], gt[0] + gt[1] * data.shape[1], data.shape[1])
y = numpy.linspace(gt[3], gt[3] + gt[5] * data.shape[0], data.shape[0])
xx, yy = numpy.meshgrid(x, y)
cs = m.contourf(xx,yy,data,range(0, 3600, 200))
for contour in cs.collections:
contour.set_clip_path(clip)
plt.show()
 I used the pyshp library for reading the shapefile, since Fiona and GDAL don't work well together, and OGR was longer
 Lines 14 to 27 create the path. A Matplotlib path is made by two arrays. One with the points (called vertices in the script), and the other with the functions for every point (called codes)
 In our case, only straight lines have to be used, so there will be a MOVETO to indicate the beginning of the polygon, many LINETO to create the segments and one CLOSEPOLY for closing it
 Of course, only the polygon for Andorra has to be used. I get it from the shapefile attributes
 The prt array is for managing multipolygons, which is not the case, but the code will create correct clipping for multipolygons
 The path is created using the Path function, and then added to a PathPatch, to be able to use it as a closed polygon. Note the trasnform=ax.transData attribute. This assumes the polygon coordinates to be the ones used in the data (longitudes and latitudes in our case). More information here
 Next code lines draw the map as usual. I have used a latlon projection, so all the values for the raster and shapefile can be used directly. If the output raster was in an other projection, the shapefile coordinates should be appended to the path using the output projection (m(pts[j][0], pts[j][1]))
 The x and y coordinates are calculated from the GDAL geotransform, and then turned into a matrix using meshgrid
 The clipping itself is made in the lines 48 and 49. For each drawn element, the method set_clip_path is applied
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