If by "use the battery" you mean "can the battery be in the circuit?" then yes.
If by "use the battery" you mean "draw power from the battery while charging it?" then no. Charging is, by definition, putting power into the battery. This is the opposite of drawing power from it. You can't do both, by definition.
Under normal operation, where the battery is powering a load, conventional current flows inside the battery from (-) to (+), through the load, and back to the battery. If something else (like your solar panel) can apply a voltage higher than the battery, then the battery becomes "the load", and current will flow through it in the other direction, reversing the redox reaction inside it, storing electric energy from the solar panel as chemical energy in the battery.
Of course, you have to make sure this current is within the operating parameters of the battery. If you charge a battery too fast, or too much, with those limits depending on the particular battery and specified in the datasheet, it will be damaged, destroyed, explode, or otherwise bad things will happen. Don't do that.
You also want to make sure that when there is not enough sun on the solar panel, and thus its voltage is lower than the battery, that the solar panel doesn't become the load, with the battery driving a (potentially very large and destructive) current through the panel. Usually this accomplished with a series diode.
There are, of course, no shortage of commercial products designed to charge batteries with solar panels. It's basically the most common thing to do with solar panels. Such a device will take care of monitoring the charge on the batteries, being sure to not damage them, preventing reverse current, etc. Some are sophisticated enough to adjust the operating point of the panels for maximum efficiency and other such neat stuff.