Multipath is an important source of error when using global navigation satellite systems (GNSS) in urban environment, leading to biased measurements and thus to false positions. This paper treats the GNSS navigation problem as the resolution of an overdetermined system, which depends on the receiver's position, velocity, clock bias, clock drift, and possible biases affecting GNSS measurements. We investigate a sparse estimation method combined with an extended Kalman filter to solve the navigation problem and estimate the multipath biases. The proposed sparse estimation method assumes that only a part of the satellites are affected by multipath, i.e., that the unknown bias vector is sparse in the sense that several of its components are equal to zero. The natural way of enforcing sparsity is to introduce an ℓ1 regularization ensuring that the bias vector has zero components. This leads to a least absolute shrinkage and selection operator (LASSO) problem, which is solved using a reweighted-ℓ1 algorithm. The weighting matrix of this algorithm is defined as functions of the carrier to noise density ratios and elevations of the different satellites. Moreover, the smooth variations of multipath biases versus time are enforced using a regularization based on total variation. For estimating the noise covariance matrix, we use an iterative reweighted least squares strategy based on the so-called Danish method. The performance of the proposed method is assessed via several simulations conducted on different real datasets.