There are areas of land that are in fact, below sea level. Are there any hills or mountains that have bases that start in these below-sea-level areas of land?
It's difficult, because to truly have a base below sea level the hill or mountain needs to be surrounded by locations below sea level on all sides. There aren't many spots on Earth's surface away from oceans below sea level at all... and many of those tend to fill with water as well. And the larger the terrain rise of the hill/mountain, generally the larger the footprint of the base, so harder and harder to get it entirely below sea level.
But topographic-map.com is one of my favorite sites for exploring topography, and its color scale works out helpful at finding areas below sea level. And its flexibility lets you root around those areas further... places like California, Egypt, Algeria, Kazakhstan, and the Caspian Sea region.
And doing so, I came across this hill in the region of ancient Israel/the modern borders of Palestine near the An-Nabi Musa area a bit northwest of the Dead Sea:
Looking at the scale and clicking around the "peak", the bases appear to be around -20 meters on all sides, and the peak elevation 74 meters, so it's just a modest 300 foot hill. But it looks to fit your request.
Google Maps also has topography lines for the location as well... which seems to roughly confirm the existence of this hill and the one further northwest, and that their bases are basically below sea level. The Layers and Street View of Google also allow you to look further at the area... there's a picture taken near one of the hills.
Such hills are oddities because of the incredibly small area of the world to be below sea level (and not covered by water). But there's nothing else at all to discourage such numerical novelties. And such areas see so little rain and are entirely disconnected from the ocean that they truly know nothing of what the rest of the world's sea level even means.
Mauna Kea is a good example of this. It is generally considered as the tallest mountain on Earth. Even if its elevation is only 4205 meters above sea level, if you add the 6000 meters of its below sea level base, it reaches more than 10 km! From USGS "How big are the Hawaiian volcanoes?":
Mauna Kea Volcano rises 13,796 feet (4,205 meters) above sea level but extends about 19,700 feet (6,000 meters) below sea level to meet the deep ocean floor. Its total height is nearly 33,500 feet (10,211 meters), considerably higher than the height of the tallest mountain on land, Mount Everest (Chomolungma in Tibetan) in the Himalayas, which is 29,029 feet (8,848 meters) above sea level.
Basically, any volcano built on an oceanic island (e.g., Piton de la Fournaise at La Réunion, Karthala at Grande Comore...) had its roots below sea level.
Following @JeopardyTempest's advice of rooting around in the Caspian Sea region, a notable feature is Mount Arlan 1,880m
As a free-standing peak in the Caspian basin, it's a proper ultra-prominent mountain indeed. While arguments could go either way on whether the nearby Caspian Sea and Garabogazköl are "properly" the base of the mountain, it's col is sadly slightly above sea level.
In The Netherlands, after the construction of the Noordoostpolder, the former islands of Schokland and Urk became hills with bases below sea level, although Urk is only partially surrounded by land, Schokland is entirely. Whereas the surrounding polder has an elevation of between -2.7 and -4.1 metre, the summit of the hill reaches as high as +0.7 metre according to the official topographic map (don't forget your high altitude gear), but it is apparently sinking and now only 0.0 metre above sea level. The former island is a UNESCO World Heritage Site and the primary (the only?) tourist attraction in the region (although there reportedly exists a nettle theme park somewhere in the region).
Schokland seen from above. Source: Wikimedia Commons.
There are also many terpen that are above sea level whereas their base is below sea level, but unlike Schokland and Urk, those are artificial hills, and if they count, the list of eligible hills in The Netherlands (in particular in Holland) becomes quite long.
I had made a mask of below sea level pixels of the Badwater Basin in this answer. The mask have a few small areas of black pixels (i.e., above sea level pixels) completely surrounded by white pixels (below sea level):
I checked on the site provided by @JeopardyTempest, there are indeed a few hills in the area. The largest one is a flat, ~30 m high hill just South of the Salt Creek interpretative trail:
Looking at it in Google Earth, I believe it is the hill (red ellipse, the red square is just a landmark for reference) pictured on this NPS photograph (public domain):
UPDATE: Ok I think I have found the best candidate, I'm amazed there actually is a true mountain completely surrounded by land below sea level, whose topographic prominence is therefore higher than its actual elevation. It's a volcano in Ethiopia, Ale Bagu: (https://www.peakbagger.com/peak.aspx?pid=77241), there is very limited information on wikipedia for such a unique mountain (https://en.wikipedia.org/wiki/Ale_Bagu)
Prior comment: Just to add to the above, I happened to be carrying out the same investigation on topographic maps, and then decided to look up online and came across this thread. A couple of other relevant locations: There seems to be a bunch around the Jordan valley, with some protruding above sea level, as the OP requested, e.g. https://en.wikipedia.org/wiki/Masada, as well as some that remain below sea level, but are perhaps a couple of hundred feet prominence above the surroundings, notably Tel Kazir by the sea of Galilee. Also a notable mention of the Salton Buttes at the Salton sea (another below sea level location in California): https://en.wikipedia.org/wiki/Salton_Buttes
The Aghzibir mud volcano (Azeri Wikipedia) appears to rise 375 feet above sea level, with its base about 85 feet below sea level, for a prominence of about 460 feet or about 140 meters. It is located about 18 kilometers south of the city of Älät (Ələt).
There are apparently a significant number of mud volcanoes in Azerbaijan (further information in Azeri), and perhaps five or six of them of them rise in the region of the country that is otherwise below sea level. Aghzibir is the tallest of these, at least according to the data from topographic-map.com.
Death Valley in California, USA reaches a depth of 282 ft (86 m) below sea level. It contains a number of hills within the valley and mountains within 1 km of the valley.
To meet your qualification, here is Shore Line Butte (188 m) rising from land that is 124 ft below sea level. Note also Cinder Hill, whose "peak" is 29 m below sea level!
Map from topographic-map.com:
This is in part a philosophical question but one that connects to the composition of continental and oceanic crust. If we consider hills and mountains we need to consider to what we are referencing as the "zero" level. Traditinally this has been sea level so any elevation is considered in terms of "meters above mean sea level" (m a.s.l.). With ongoing climate change sea, levels are rising so the elevations are essentially decrasing we need to reference some reference level if qwe need to express elevations. I think it is still an open question as to what this datum will be. In the geological past sea levels have changed depending on a number of factors such as the rate of renewal of new oceanic plate (being relatively low density due to its inherent heat), the volume of land locked ice sheets (lowering the sea level and volume) and other factors.
The question can only be answered given the conditions under which the areas considered have been formed, i.e. the formation of new warm and therefore light crust and the volume of land-locked ice under a given climate scenario. The current distribution of elevation above and below sea level is not necessarily in synch with our current climate and certainly current climate change. Therefore there is much to consider when interpreting current topography in terms of current climate and certainly current climate change.